K-Theory of Azumaya Algebras

K -Theory of Azuma y a Algebras A thesis for th e d e gr e e of Do ctor of Philosophy submitte d to the Scho ol of Mathemati cs and Physics of Que en ’s University Belfast. Judith Ruth Millar MSci September 2010 Ac kno wledgemen ts I w ould lik e to ex press m y gratitude to m y s up ervisor D r . Roozb eh Hazrat for his a d- vice, encouragemen t, patience and the many hours of his time o v er the last four y ears. I w ould also lik e to thank the mem b ers of the Department of P ure Mathematics, and esp ecially the p ostgraduate studen ts I ha v e met here at Queen’s. I a c kno wledge the ﬁnancial assistance I receiv ed from the Departmen t of Education and Learning. Finally , I w o uld lik e to thank my family for their support throug ho ut m y univ ersit y life. Con ten ts Notation 3 0 In tro duction 5 1 Azuma y a Algebras 12 1.1 F aithfully pro jectiv e mo dules . . . . . . . . . . . . . . . . . . . . . . 12 1.2 Separable algebras o ve r comm utativ e rings . . . . . . . . . . . . . . . 18 1.3 Other deﬁnitions of separabilit y . . . . . . . . . . . . . . . . . . . . . 24 1.4 Azuma y a algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5 F urther c haracterisations o f Azuma y a algebras . . . . . . . . . . . . . 29 1.6 The dev elopmen t of the theory of Azuma ya algebras . . . . . . . . . . 35 2 Algebraic K -Theory 37 2.1 Lo w er K - groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2 Lo w er K - groups of cen tral simple algebras . . . . . . . . . . . . . . . 44 2.3 Higher K -Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3 K -Theory of Azuma ya Algebras 55 3.1 D -functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2 Homology of Azuma ya algebras . . . . . . . . . . . . . . . . . . . . . 61 4 Graded Azuma y a Algebras 63 4.1 Graded rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2 Graded mo dules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 1 Con ten ts 2 4.3 Graded cen tr a l simple alg ebras . . . . . . . . . . . . . . . . . . . . . 76 4.4 Graded matrix rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.5 Graded pro jectiv e mo dules . . . . . . . . . . . . . . . . . . . . . . . . 93 5 Graded K -Theory of Azuma ya A lgebras 99 5.1 Graded K 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2 Graded K 0 of strongly graded rings . . . . . . . . . . . . . . . . . . . 105 5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 09 5.4 Graded D -functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 6 Additiv e Comm utators 119 6.1 Homogeneous additiv e comm utato rs . . . . . . . . . . . . . . . . . . . 120 6.2 Graded splitting ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.3 Some results in the non-graded setting . . . . . . . . . . . . . . . . . 131 6.4 Quotien t division ring s . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Bibliograph y 138 Index 145 Notation Throughout all rings are assumed to b e a sso ciativ e and any ring R has a m ultiplica- tiv e iden tity elemen t 1 R . A subring S of R con tains the iden tity elemen t of R . W e assume that a ring homomorphism R → R ′ tak es the identit y of R to the iden tity of R ′ . Mo dules o v er a ring R are assume d to b e left R -mo dules unless otherwise stated. F or an R - mo dule M , w e a ssume that 1 R m = m for ev ery m ∈ M . F or a homo- morphism b et w een R -mo dules, w e will use the terms R -mo dule homomorphism and R -linear homomorphism interc hangeably . F or a ﬁeld F , a division algebra D ov er F is deﬁned to b e a division ring with cen tre F s uc h that [ D : F ] < ∞ . Some sym b ols used F or a ring R , R ∗ the group of units o f R ; that is, t he elemen ts of R whic h ha ve a m ultiplicativ e inv erse; R - M od the category o f R -mo dules with R -mo dule homomorphisms betw een them; P r( R ) the subcategory of R - M od cons isting of ﬁnitely generated pro jectiv e R -mo dules. F or a m ultiplicativ e group G and x, y ∈ G , 3 Notation 4 [ x, y ] the comm utator xy x − 1 y − 1 ; [ G, G ]= G ′ the c ommutator sub gr o up of G ; that is, the (normal) subgro up of G generated b y the commutators; G/G ′ the ab elian i s a tion of G . Chapter 0 In tro duction F or a division algebra D ﬁnite dimensional o v er its cen tre F , consider the m ultiplica- tiv e g roup D ∗ = D \ 0. The structure of subgroups of D ∗ is not know n in general. In 1953, Herstein [ 35 ] prov ed that if D is of c hara cteristic p 6 = 0, t hen ev ery ﬁnite subgroup of D ∗ is cyclic. This is an easy result in the setting of ﬁelds (see [ 37 , Thm. V.5.3]), so the ﬁnite subgroups of suc h division algebras b ehav e in a similar w a y to those of ﬁelds. W e giv e an example here to show that this result do esn’t necessarily hold for division algebras of characteristic zero. Hamilton’s quaternions H form a division algebra o f c haracteristic zero, but the subgroup {± 1 , ± i, ± j, ± k } is a ﬁnite subgroup of H ∗ whic h is no t ab elian, so not cyclic. In 1955 , Amitsur classiﬁed all ﬁnite subgroups o f D ∗ in his inﬂuen tial pap er [ 1 ]. Since then , subgroups of the group D ∗ ha v e b een studied b y a num b er of p eople (see for example [ 23 , 25 , 42 , 46 ]). Herstein [ 36 ] show ed that a non-central elemen t in a division algebra has inﬁnitely many conjugates. Since normal subgroups are inv aria n t under conjugation, this show s that non-cen tral norma l subgro ups are “ big” in D ∗ . Note that if D is a non-comm utativ e division algebra, then non- t r ivial non- cen tral normal subgroups exist in D ∗ . F or example D ′ , the subgroup of D ∗ generated by the m ultiplicative commutators, is non-trivial and is a non-cen tral normal subgroup of D ∗ . So as for normal subgroups, w e could ask: how large are maximal subgroups of 5 Chapter 0. In tr o duction 6 D ∗ ? But it remains as a n op en question whether ma ximal subgroups ev en exist in D ∗ . The existence of maximal subgroups of D ∗ is connected with the non- trivialit y of K -group CK 1 ( D ). F o r a division algebra D with cen tre F , w e not e that CK 1 ( D ) ∼ = D ∗ / ( F ∗ D ′ ). The group CK 1 ( D ) is related to algebraic K -theory; more sp eciﬁcally , to the functor K 1 . Before discussing the group, w e indicate ho w algebraic K -theory has dev elop ed. Algebraic K -t heory deﬁnes a sequence of functors K i from the categor y of rings to the category of ab elian gro ups. F o r the low er K -groups, the functor K 0 w as in tro duced in the mid-19 50s by G rothendiec k, and the functor K 1 w as dev elop ed in the 1960 s b y Ba ss. Man y attempts were made to extend these functors to cov er all K i for i ≥ 0. Milnor deﬁn ed the functor K 2 in the 1 960s, but it w as not clear ho w to construct the higher K -functors. The functor K 2 is deﬁned in suc h a w ay that there is an exact se quence linking it with K 0 and K 1 (see [ 58 , Thm. 4.3.1]). The “correct” deﬁnition of the higher K -functors w as required not only to prov ide suc h a n exact sequence connecting the functors, but also to co ve r the giv en deﬁnitions of K 0 and K 1 . Then in 1 974, Quillen g av e t wo diﬀeren t constructions of the higher K -functors, whic h are equiv a lent for rings and whic h satisfy these exp ected prop erties (see [ 58 , Ch. 5]). It is straigh tforw ard to describ e the low er K -gro ups concretely . The group K 0 can b e considered as the group completion of the monoid of isomorphism classes of ﬁnitely generated pro jectiv e mo dules, and K 1 is the ab elianisation of the inﬁnite general linear group (see Chapter 2 for t he details). The higher K -groups are considerably more diﬃcult to compute. They are, ho we ve r, functorial in construction. Returning to the setting of division algebras, consider a cen tra l simple algebra A . Then by the Artin-W edderburn Theorem, A is isomorphic to a matrix M n ( D ) o v er a division a lgebra D . Let F b e the cen tre of D . Since eac h K i , i ≥ 0, is a functor from the category of rings to the category of ab elian g r oups, the inclusion map F → A induces a map K i ( F ) → K i ( A ). Let ZK i ( A ) denote the k ernel of this Chapter 0. In tr o duction 7 map and CK i ( A ) denote the cok ernel. This gives an exact sequence 1 → ZK i ( A ) → K i ( F ) → K i ( A ) → CK i ( A ) → 1 . (1) So the group CK 1 ( A ) is deﬁned to b e cok er  K 1 ( F ) → K 1 ( A )  . Then it can b e sho wn that CK 1 ( A ) is isomorphic to D ∗ /F ∗ n D ′ , and it is a b ounded torsion ab elian group (see Section 2.2 for the details). F or a division algebra D , if CK 1  M n ( D )  is not the trivial group f o r some n ∈ N , then D ∗ has a normal maximal subgroup (see [ 33 , Section 2]). It has b een conjectured that if CK 1 ( D ) is trivial, then D is a quaternion algebra (see [ 30 , p. 4 0 8]). One of the most signiﬁcant results in this direction pro v es that if D is a division algebra with cen tre F suc h that D ∗ has no ma ximal subgroups, then D and F satisfy a num b er of conditions (see [ 33 , Thm. 1]). This result ensures tha t certain division algebras, for example division algebras of degree 2 n or 3 n for n ≥ 1, hav e maximal subgroups. W e note tha t CK 1 can b e considere d as a functor from the category of cen tral simple algebras o v er a ﬁxed ﬁeld t o the category of ab elian groups (see Section 2.2 ). In fa ct, t his can b e generalised to co v er comm utativ e rings. Cen tra l simple a lg ebras o v er ﬁelds are generalised by Azuma y a algebras ov er comm ut a tiv e rings. Azumay a algebras w ere originally deﬁned as “prop er maxim ally cen tral algebras” b y Azuma y a in his 1951 pap er [ 3 ]. W e outline in Section 1.6 how the deﬁnition ha s dev elop ed since then. An Azuma y a algebra A o v er a commutativ e ring R can b e deﬁned a s an R -algebra A suc h that A is ﬁnite ly generated as an R -mo dule and A/ m A is a ce ntral simple R/ m-algebra for all m ∈ Max( R ) (see Theorem 1.5.3 for some equiv alen t deﬁnitions). Then CK 1 can also b e considered as a functor from the category of Azuma y a al- gebras ov er a ﬁxed commu tat iv e ring to the category of a b elian gr o ups (see pag e 58 ). Related to this, v arious CK 1 -lik e functors on t he categories of ce ntral simple algebras and Azuma ya alg ebras ha v e b een inv estigated in [ 27 , 28 , 45 ]. In these pap ers v arious abstract f unctors hav e b een deﬁned, which hav e similar prop erties to the functor CK 1 . F or example, in [ 28 ], the functor deﬁned there is used to sho w that the K - Chapter 0. In tr o duction 8 theory of an Azuma y a algebra ov er a lo cal ring is a lmo st the same a s the K -theory of the base ring. Along the same lines, in Chapter 3 w e deﬁne a n abstract functor, called a D - functor, whic h also has similar prop erties to CK 1 . W e sho w that the range o f a D -functor is a b ounded torsion ab elian group, and that CK i and ZK i , for i ≥ 0, are D -functors. By com bining these results with ( 1 ), w e sho w in Theorem 3.1.5 that if A is an Azuma ya algebra free ov er its cen tre R of rank n , then K i ( A ) ⊗ Z [1 /n ] ∼ = K i ( R ) ⊗ Z [1 /n ] for an y i ≥ 0. This allo ws us to extend the results o f [ 28 ] to co v er Azuma y a algebras o v er semi-lo cal rings (see Corollary 3.1.7 ). Th us f ar, w e ha ve b een considering divis ion algebras. R ecen t ly Tignol and W adsw orth [ 67 , 66 ] hav e studied division algebras equipped with a v aluation. V al- uations are more common on ﬁelds than on division algebras. Ho w ev er they noted that a num b er o f division algebras are equipped with a v aluation, and the v aluation structure on the division a lg ebra con tains a signiﬁcan t amoun t of inf ormation ab o ut the division algebra. A division algebra D equipped with a v alua t ion giv es rise to an asso ciated graded division algebra gr( D ). These graded division algebras ha v e b een studied in [ 6 , 34 , 38 , 39 , 67 ]. In these pap ers, as they are considering graded division algebras asso ciated to division algebras with v aluat ions, their gra de groups are totally ordered ab elian groups. It w as noted in [ 39 ] that it is relative ly easier to work with graded division algebras, and that not mu ch information is lost in passing b etw een the graded a nd non-graded settings. W e sho w in Theorem 4.3.3 that a graded cen tral simple algebra (so, in particular, a graded division algebra) with an ab elian grade group is an Azuma y a a lg ebra, and therefore t he results of Chapter 3 also hold in this setting. But in the graded setting, w e can also consider graded ﬁnitely generated pro jectiv e mo dules ov er a giv en graded ring. W e de ﬁne the graded K - t heory of a graded ring R to b e K gr i ( R ) = K i ( P gr( R )), Chapter 0. In tr o duction 9 where P gr( R ) is the category of graded ﬁnitely generated pro jectiv e R -mo dules. Ho w ev er, c onsidering gra ded K -theory of graded Azuma ya algebras, Example 5.3.2 giv es a gr a ded Azuma ya algebra such that its graded K -theory is not isomorphic to the graded K - theory o f its cen tre. In t his example, w e tak e the real quater- nion algebra H . Then w e sho w H is a graded Azuma y a algebra o v er R , with K gr 0 ( H ) ⊗ Z [1 /n ] ∼ = Z ⊗ Z [1 /n ] and K gr 0  Z ( H )  ⊗ Z [1 /n ] ∼ = ( Z ⊕ Z ⊕ Z ⊕ Z ) ⊗ Z [1 /n ], so they are not isomorphic. Th us the results of Chapter 3 do not follow immediately in the setting of graded K -theory . But for a graded Azuma y a algebra su b j ect to certain conditions, w e show in The- orem 5.4.4 that its graded K -theory is almost the same as the graded K -theory of its cen tre. More precisely , for a comm utativ e graded ring R , w e let Γ ∗ M n ( R ) denote the elemen ts ( d ) ∈ Γ n suc h t ha t G L n ( R )[ d ] 6 = ∅ , where GL n ( R )[ d ] are in v ertible n × n matrices with “shifting” (see page 71 ). W e sh ow that if A is a graded Azumay a alge- bra whic h is graded free o v er its cen tre R of rank n , suc h that A has a homogeneous basis with degrees ( δ 1 , . . . , δ n ) in Γ ∗ M n ( R ) , then for an y i ≥ 0, K gr i ( A ) ⊗ Z [1 /n ] ∼ = K gr i ( R ) ⊗ Z [1 /n ] . Another K -g roup whic h has b een s tudied in the setting o f division algebras is the reduced Whitehead group SK 1 (see for example [ 53 ]). F or a division algebra D , the group SK 1 ( D ) is deﬁned to b e D (1) /D ′ where D (1) is the k ernel of the reduced norm and D ′ is the group generated b y the m ultiplicativ e comm utator s of D . F o r a graded division a lgebra D , it has b een sho wn that SK 1 ( QD ) ∼ = SK 1 ( D ), where QD is the quotien t division ring of D (see [ 34 , Thm. 5.7]). R elated t o this, we study additiv e comm utators in the setting of graded division algebras in Chapter 6 . W e show t ha t for a graded division algebra o ve r its cen tre F , whic h is No etherian as a ring, then D [ D , D ] ⊗ F QF ∼ = QD [ QD , QD ] . Chapter 0. In tr o duction 10 Summary of the Thesis In Chapter 1 , w e com bine v arious results fr o m t he literature to sho w some of the deﬁnitions of an Azuma y a algebra, their basic prop erties and the equiv alence of some of these deﬁnitions. W e also outline how the deﬁnition has prog r essed since the w o r k of Azuma y a. W e note here that Gro thendiec k [ 26 , § 5] also deﬁnes an Azumay a algebra o n a sc heme X with structure sheaf O X , but w e do not consider tha t p oint of view. In Chapter 2 , w e b egin by recalling the deﬁnitions of the lo w er K -groups K 0 , K 1 and K 2 . W e then lo ok a t the low er K -groups of cen tral simple algebras, including the f unctors CK 0 and CK 1 , and some of their prop erties. W e ﬁnish the c hapter by recalling some prop erties of the higher K - groups. In Chapter 3 , we deﬁne an abstract functor, called a D -f unctor, deﬁned on the category of Azuma y a algebras o ve r a ﬁxed comm utativ e ring. This allo ws us to s how that the K -theory of an Azuma y a algebra free ov er its cen tre is almost the same as the K -theory of its cen tre. W e also note that Corti ˜ nas a nd W eib el [ 13 ] ha ve s hown a similar result fo r the Ho c hsc hild homology of an Azuma y a algebra, whic h w e men tion in this c hapter. Chapter 4 introduces graded ob jects. O ften in the literature the grade groups are ab elian and totally ordered, so torsion-free. W e b egin this section by adopting in the gr a ded setting some theorems that w e require from the non-graded setting. Some of these results hold for gra de groups whic h are neither ab elian nor totally ordered. Though in some cases w e require additional conditio ns on the grade group. W e sho w that for a graded division ring D graded by a n ar bit r a ry group, a graded mo dule o ve r D is gr a ded free and has a uniquely deﬁned dimension. F or a graded ﬁeld R a nd a graded cen tral simple R -algebra A g r a ded b y an ab elian group, w e sho w that A is a graded Azuma ya algebra ov er R . W e also pro v e a num b er of results for graded matrix rings graded b y a r bitr ary groups. W e b egin Chapter 5 by deﬁning the group K 0 in t he setting of graded ring s. W e sho w what this group lo oks lik e for a trivially graded ﬁeld and for a strongly g raded Chapter 0. In tr o duction 11 ring. F o r a sp eciﬁc example of a gra ded Azuma y a algebra, we show that its graded K -theory is not the same as its usual K -theory (see Example 5.3.5 ). Then in a similar w ay to Chapter 3 , w e deﬁne an abstract functor called a graded D -functor. This allows us to pro v e that the graded K -theory of a graded Azuma y a a lg ebra (sub ject to some conditions) is almost the same a s the graded K -t heory of its c entre. In Chapter 6 , w e study additiv e comm utat ors in t he setting of graded division algebras. W e observ e in Section 6.2 that the reduced trace holds in this setting. W e then recall the deﬁnition of the quotien t division a lgebra, and sho w in Coro llary 6.4.5 ho w the subgroup generated by homogeneous additive comm utator s in a graded division algebra relates to that of the quotien t division algebra. Chapter 1 Azuma y a Algebras The concept of an Azuma y a algebra o ve r a c ommutativ e ring generalises the conce pt of a cen t ral simple algebra ov er a ﬁeld. The term Azumay a algebra originates from the work done b y Azuma y a in his 1951 pap er “On maximally central a lgebras” [ 3 ]. The deﬁnition has dev elop ed since then, and w e will o utline in Section 1.6 ho w it has progressed. In Theorem 1.5.3 , w e state a n um b er o f equiv alent reform ulations of this deﬁnition. This chapter is organised as follow s. W e b egin this c hapter by recalling t he v arious deﬁnitions o f the term “faithfully pro jectiv e”, whic h ar e required fo r the deﬁnition of an Azuma y a algebra (see Deﬁnition 1.4.1 ). In Sections 1 .2 and 1.3 w e discuss separable alg ebras, whic h can also b e used to deﬁne Azuma ya algebras. The deﬁnition of an Azuma y a algebra in stated in Section 1.4 , along with some exam- ples and pr o p erties, and in Section 1.5 w e sho w some a dditional c haracterisations of Azuma y a algebras. W e conclude this c hapter by summarising some of the k ey progressions in the dev elopmen t of the theory of Azuma y a algebras. 1.1 F aithfully pro jectiv e mo dules Let R b e a (p ossibly non-commutativ e) r ing . Consider a co v ariant additiv e functor T from the category of (left or right) R -mo dules to some categor y o f mo dules. W e 12 Chapter 1. Azuma ya Algebras 13 sa y that T is an exa c t functor if, whenev er L → M → N is an exact sequence of R -mo dules, T ( L ) → T ( M ) → T ( N ) is exact. F urther T is deﬁned to b e a faithful ly exact functor if the sequence T ( L ) → T ( M ) → T ( N ) is exact if and only if the sequence L → M → N is exact. W e recall that an R - mo dule M is called faithful if r M = 0 implies r = 0 or, equiv alen tly , if its annihilator Ann ( M ) = { x ∈ R : xm = 0 fo r all m ∈ M } is zero. An R - mo dule M is called a ﬂat mo dule if the functor − ⊗ R M is an exact functor from the category of r ig h t R -mo dules to the catego ry o f ab elian groups. An R -mo dule P is called a p r oje ctive mo dule if the functor Hom R ( P , − ) is an exact functor from the category of left R -mo dules to the category of ab elian groups. This is equiv a len t to sa ying that P is a direct summand of a free R -mo dule. If P is a pro jectiv e R -mo dule whic h is ﬁnitely generated b y n elemen ts, then P is a direct summand of R n . See Magurn [ 48 , Ch. 2] for results in v olving pro jectiv e mo dules. The follow ing results o n faithfully exact func tors are from Ishik aw a [ 40 , p. 30–33]. Theorem 1.1.1. L et T b e an exact functor fr om the c ate gory of left (r es p. right) R -mo dules to some c a te gory of mo dules. Th en the fol lowing ar e e quivalen t: 1. T is a fa ithful ly exact functor. 2. T ( A ) 6 = 0 for every non zer o l e ft (r e sp. right) R -mo dule A . 3. T ( φ ) 6 = 0 fo r every non zer o R -line ar hom o morphism φ . 4. T ( R/I ) 6 = 0 for every pr op er left (r esp. right) ide al I of R . 5. T ( R/ m) 6 = 0 for every maximal left (r esp. right) i de al m of R . W e will consider the left v ersion of this theorem in the pro of b elo w. The right v ersion follows analog ously . Pro of. (1) ⇒ (2): Let T ( A ) = 0 fo r an R -mo dule A . Then since T (0) is the zero mo dule, T (0) → T ( A ) → T (0) is exact. By (1), this implies that 0 → A → 0 is Chapter 1. Azuma ya Algebras 14 exact, pro ving A = 0. (2) ⇒ (3): Let φ : X → Y b e an R -linear homomorphism. Then we hav e exact sequence s X φ ′ → Im( φ ) → 0 and 0 → Im ( φ ) i → Y , where i is t he inclus ion map. Since T is an exact functor, we get the follo wing comm utativ e diagr a m with its row and column exact: 0   T ( X ) T ( φ ′ ) / / T ( φ ) % % K K K K K K K K K K T  Im( φ )  T ( i )   / / 0 T ( Y ) If T ( φ ) = 0, then T ( i ) ◦ T ( φ ′ ) = 0. This implies that Im  T ( φ ′ )  ⊆ ker  T ( i )  = 0, so T ( φ ′ ) = 0 and, since T ( φ ′ ) is surjectiv e, T  Im( φ )  = 0. By condition (2), Im( φ ) = 0, so φ = 0. (3) ⇒ (1) : Let T ( A ) T ( f ) − → T ( B ) T ( g ) − → T ( C ) b e exact. Since T ( g ◦ f ) = T ( g ) ◦ T ( f ) = 0, by condition (3 ) , g ◦ f = 0 and Im( f ) ⊆ ker( g ). W e ha ve exact sequences 0 → k er( g ) j → B g → C , A f ′ → Im( f ) → 0 a nd 0 → Im( f ) i → k er( g ) p → k er( g ) / Im( f ) → 0, where i and j are inclusion maps. Since T is an exact functor, we obtain the f o llo wing comm utativ e dia gram with exact ro ws a nd columns: 0   0 T  Im( f )  o o T ( i )   T ( A ) T ( f )   T ( f ′ ) o o 0 / / T  k er( g )  T ( j ) / / T ( p )   T ( B ) T ( g ) / / T ( C ) T  k er( g ) / Im( f )    0 F or x ∈ T  k er ( g )  ,  T ( g ) ◦ T ( j )  ( x ) = 0, so T ( j )( x ) ∈ k er  T ( g )  = Im  T ( f )  . Chapter 1. Azuma ya Algebras 15 So there is an eleme nt y ∈ T ( A ) suc h that T ( f ) ( y ) = T ( j )( x ). Hence T ( j )( x ) = T ( f )( y ) = T ( j ) ◦ T ( i ) ◦ T ( f ′ )( y ), so that x =  T ( i ) ◦ T ( f ′ )  ( y ), since T ( j ) is injectiv e. This show s that T ( i ) is surjectiv e, and hence T  Im( f )  ∼ = T  k er( g )  , whic h means that T ( p ) = 0. By condition (3), this implies p = 0, so Im( f ) = k er( g ), pro ving A f → B g → C is exact. (2) ⇒ (4) and (4) ⇒ (5) are trivial. (5) ⇒ (2): Let T ( A ) = 0. Let a ∈ A and let R a b e the left R -submo dule of A generated by a . Since 0 → R a → A is exact a nd T is exact, T ( Ra ) = 0. Let L ( a ) = { r ∈ R : r a = 0 } , which is a left ideal of R . If L ( a ) 6 = R , then there is a maximal ideal m of R con taining L ( a ). W e hav e an exact sequ ence R/ L ( a ) → R/ m → 0. Since R → Ra is surjectiv e, R a ∼ = R/ L ( a ) b y the Fir st Isomorphism Theorem, and w e ha v e an exact sequence 0 = T ( Ra ) ∼ = T  R/ L ( a )  → T ( R/ m) → 0. This implies T ( R/ m) = 0, con tradicting (5). So L ( a ) = R , whic h implies a = 0 and therefore A = 0.  F or ﬁxed left R - mo dules P and M , the functors T ( − ) = Hom R ( P , − ) and U ( − ) = − ⊗ R M are cov ariant functors deﬁned on the category of left R -mo dules and right R -mo dules, r esp ective ly . Deﬁnition 1.1.2. An R -mo dule P is said to b e fa i thful ly pr oje c tive if T ( − ) = Hom R ( P , − ) is a fait hf ully exact functor and an R -mo dule M is said to b e faithful ly ﬂat if U ( − ) = − ⊗ R M is a fa ithfully exact functor. By a pplying Theorem 1.1.1 to the functors U and T respectiv ely , w e get the follo wing theorems. Theorem 1.1.3. L et M b e a ﬂat l e f t R -mo dule. Then the fol lowing ar e e quivalen t: 1. M is faithful ly ﬂat. 2. A ⊗ R M 6 = 0 for e v e ry non zer o righ t R -mo d ule A . 3. φ ⊗ R id M 6 = 0 for every non zer o right R -lin e ar homomorphism φ . Chapter 1. Azuma ya Algebras 16 4. I M 6 = M for every pr op er right ide al I o f R . 5. m M 6 = M for every maximal right ide al m of R . Pro of. F o llows immediately from Theorem 1.1.1 . Note that in par t (4), R/I ⊗ R M ∼ = M /I M , and M /I M = 0 if and only if M = I M .  Theorem 1.1.4. L et P b e a p r oje ctive left R -mo d ule. Then the fol low ing ar e e quiv- alent: 1. P is fai thful ly pr oje ctive. 2. Hom R ( P , A ) 6 = 0 for every non zer o left R - m o dule A . 3. Hom R ( P , φ ) 6 = 0 for every non zer o left R -line ar homomorph ism φ . 4. Hom R ( P , R/I ) 6 = 0 for every pr op er left ide al I of R . 5. Hom R ( P , R/ m) 6 = 0 for eve ry max i m al left ide al m of R . Pro of. F o llows immediately from Theorem 1.1.1 . In part ( 3 ), if φ ∈ Hom R ( X , Y ), then T ( φ ) = Hom R ( P , φ ) : Hom R ( P , X ) → Hom R ( P , Y ); ψ 7→ φ ◦ ψ .  Prop osition 1.1.5. I f an R -m o dule P is faithful ly pr oje c tive, then P is pr oje ctive and fa i thful ly ﬂat. F urther, when the ring R is c ommutative the c onverse holds. Pro of. If an R - mo dule P is fa ithfully pro j ective , it is pro jectiv e, and therefore also ﬂat (see [ 43 , Prop. 4.3]). Using Theorem 1.1.3 (2), we assume A ⊗ R P = 0 and need to pro v e t hat A = 0. Then b y [ 7 , § I I.4.1, Prop. 1], Hom R ( P , Hom Z ( A, A )) ∼ = Hom Z ( A ⊗ R P , A ) = Hom Z (0 , A ) = 0 . Since P is faithfully pro jectiv e, b y Theorem 1.1.4 (2), Hom Z ( A, A ) = 0, so A = 0 . Con v ersely , let R b e comm utativ e and P b e fait hf ully ﬂat and pro j ective . By Theorem 1 .1 .3 (5), for an y maximal ideal m of R , w e ha ve P / m P 6 = 0 . Since R → Chapter 1. Azuma ya Algebras 17 R/ m is a surjectiv e ring homomorphism, R/ m-linear maps can b e considered as R - linear maps and w e ha v e Hom R ( P / m P , R/ m) = Hom R/ m ( P / m P , R/ m). Since R/ m is a ﬁeld and P / m P 6 = 0, its dual mo dule Hom R/ m ( P / m P , R/ m) is also non-zero, using dim R/ m (Hom R/ m ( P / m P , R/ m)) ≥ dim R/ m ( P / m P ) , from [ 37 , p. 204, Remarks]. W e hav e an exact sequence 0 − → Hom R ( P / m P , R/ m) − → Hom R ( P , R/ m) , so Hom R ( P , R/ m) 6 = 0, since Hom R ( P / m P , R/ m) 6 = 0. By Theorem 1.1.4 (5), P is faithfully pro jectiv e.  In the following prop osition, w e sho w that the deﬁnition of a faithfully pro jectiv e R -mo dule (Deﬁnition 1.1.2 ) can b e expressed in a num b er of diﬀeren t w ay s, which are equiv alen t to the deﬁnition giv en ab ov e provided R is a comm utativ e ring. The second deﬁnition is from [ 4 , p. 39] or [ 22 , p. 186], and the third from [ 41 , p. 52]. Prop osition 1.1.6. L et R b e a c omm utative rin g , and l e t P b e an R -m o dule. Then the fol lowing a r e e q uiva lent: 1. P is fai thful ly pr oje ctive; 2. P is ﬁn i tely gener a te d, pr oje ctive and f a ithful as an R -mo dule; 3. P is pr oje ctive ov er R and P ⊗ R N = 0 impli e s N = 0 for any left R -mo dule N . Pro of. (1) ⇔ (3): This follows from Prop osition 1.1.5 and Theorem 1.1.3 (2). (1) ⇔ (2): See Bass [ 4 , Cor. I I.5.10 ].  W e sho w b elow ho w faithfully ﬂat mo dules a r e r elat ed to mo dules whic h are faithful and ﬂat. Chapter 1. Azuma ya Algebras 18 Prop osition 1.1.7. L et R b e a ring. A faithful ly ﬂat le ft R -mo dule M is b oth faithful and ﬂat. Pro of. See Lam [ 43 , Prop. 4.73], with minor alterations for left mo dules.  In general the con ve rse do es not hold. F or example, if R = Z , then the mo dule Q is faithful and ﬂat, since Q = ( Z \ 0) − 1 Z is a lo calisation o f Z , a nd w e kno w lo calisations are ﬂat (see [ 48 , Prop. 6.56]). But b y Theorem 1.1.3 , Q no t faithfully ﬂat o ve r Z , since for the ideal 2 Z o f Z , Q ⊗ Z ( Z / 2 Z ) = 0. Ev en a faithful a nd pro jectiv e R -mo dule M is not necessarily faithfully ﬂat ov er R . F or example, let R b e the direct pro duct Z × Z × · · · , and let M b e the ideal Z ⊕ Z ⊕ · · · in R . Then M is faithful as a left R -mo dule, a nd it is pro jectiv e (see [ 43 , Eg. 2.12 C]). But w e ha v e M 2 = M , so for a n y maximal ideal m of R con taining M , we ha v e M m = M . So b y Theorem 1.1.3 , M is not f aithfully ﬂat. 1.2 Separable alg e bras o v er comm utativ e ri ngs In this section, we let R denote a comm utative ring. Let A b e an R -algebra, and let A e = A ⊗ R A op b e the en v eloping algebra of A , where A op denotes the o pp osite algebra of A . Then the R -algebra A e has a left action on A induced b y: ( a ⊗ b ) x := axb for a, x ∈ A , b ∈ A op , whic h is denoted b y ( a ⊗ b ) ∗ x . An y A -bimo dule M can also b e view ed as a left A e -mo dule. W e set M A = { m ∈ M : ma = am fo r all a ∈ A } . There is an A e -linear map µ : A e → A ; a ⊗ b 7→ ( a ⊗ b ) ∗ 1 = ab , extended linearly , and w e let J denote the ke rnel of µ . Chapter 1. Azuma ya Algebras 19 Deﬁnition 1.2.1. An R -algebra A is said to b e sep ar able ov er R if A is pro jectiv e as a left A e -mo dule. The following t w o theorems show some equiv alen t characterisations of separabil- it y . Theorem 1.2.2. L et A b e an R -alge b r a. The fol low ing ar e e quivalent: 1. A is sep ar ab l e . 2. The e xact se quenc e of left A e -mo dules 0 − → J − → A e µ − → A − → 0 splits. 3. The f unctor ( − ) A : A e - M od → R - M od is exact. 4. Ther e is an element e ∈ A e such that e ∗ 1 = 1 and J e = 0 . 5. Ther e is an element e ∈ A e such that e ∗ 1 = 1 a n d ( a ⊗ 1) e = (1 ⊗ a ) e for al l a ∈ A . Suc h a n elemen t e as in Theorem 1.2.2 is an idemp o ten t, called a se p ar ability idemp o tent for A , since e 2 − e = ( e − 1 ⊗ 1) e ∈ J e = 0. Pro of. [ 41 , Lemma I I I.5.1.2], [ 17 , Prop. I I.1.1 ] (1) ⇔ (2): T he forw ard direction follo ws immediately fr o m the deﬁnition of a pro j ective mo dule. F or the conv erse, using kno wn results inv olving pro jectiv e mo dules (see [ 48 , Cor. 2.16]), (2) implies A e ∼ = J ⊕ A , so A is pro jectiv e. (1) ⇔ (3): F or all A -bimo dules M , t he natural map ρ M : Hom A e ( A, M ) − → M A f 7− → f (1) Chapter 1. Azuma ya Algebras 20 is an isomorphism of R -mo dules, with the in v erse b eing ρ − 1 M : M A − → Hom A e ( A, M ) x 7− → R x : A → M a 7→ ax. Since A is separable if and only if Hom A e ( A, − ) is an exact functor, this prov es the equiv alence of (1) and (3). (2) ⇒ (4): Let γ : A → A e b e an A e -mo dule homomorphism suc h that µ ◦ γ = id A . Let e = γ (1), so that 1 = µ ( e ) = e ∗ 1. T o show that J e = 0, let a ∈ J . Then as γ is A e -linear, ae = aγ (1) = γ ( a ∗ 1) = 0, since w e hav e a ∗ 1 = µ ( a ) = 0, prov ing J e = 0, as required for (4). (4) ⇒ (5): F rom (4), w e hav e an elemen t e ∈ A e suc h that e ∗ 1 = 1 and J e = 0. Let a ∈ A b e arbitrary . The n µ (1 ⊗ a − a ⊗ 1 ) = 0, so 1 ⊗ a − a ⊗ 1 ∈ J . Hence (1 ⊗ a − a ⊗ 1) e = 0; that is, (1 ⊗ a ) e = ( a ⊗ 1) e , proving (5) . (5) ⇒ (2): If e is an elemen t o f A e satisfying the conditions in (5), w e can deﬁne a map γ by γ : A → A e ; a 7→ (1 ⊗ a ) e . Using the assumption that ( a ⊗ 1) e = (1 ⊗ a ) e for all a ∈ A , we can show t ha t γ is an A e -mo dule homomor phism. It is a righ t in v erse of µ since, f or a ∈ A , writing e = P x i ⊗ y i giv es µ ◦ γ ( a ) = µ  (1 ⊗ a ) e  = µ  X ( x i ⊗ y i a )  =  X x i y i  a = 1 .a = id A ( a ) , completing the pro of.  Theorem 1.2.3. L et A b e an R -algebr a which is ﬁnitely gener ate d a s an R -mo d ule. The fo l lowing ar e e quivale nt: Chapter 1. Azuma ya Algebras 21 1. A is sep ar ab l e over R . 2. A m is sep ar a b l e over R m for al l m ∈ Max( R ) . 3. A/ m A is sep ar able ov e r R/ m for al l m ∈ Max ( R ) . Pro of. See [ 41 , Lemma I I I.5.1.10].  F or a free R -mo dule F , w e kno w that F is isomorphic to a direct sum of copies of R as a left R -mo dule; that is, F ∼ = L i ∈ I R i for R i = R (see [ 37 , Thm. IV.2.1]) . Let f i ∈ Hom R ( F , R ) b e the pro jection of R i on to R , and let e i b e the elemen t of F with 1 in the i -th p osition and zeros elsewhere . Then clearly the f o llo wing r esults hold: 1. for ev ery x ∈ F , f i ( x ) = 0 for all but a ﬁnite subset of i ∈ I ; 2. for ev ery x ∈ F , P i f i ( x ) e i = x . The following lemma sho ws t ha t w e ha v e similar results when w e consider pro jec- tiv e mo dules, rather than free mo dules. Moreov er, suc h prop erties a r e suﬃcien t to c haracterise a pro jectiv e mo dule. Lemma 1.2.4 (D ual Basis Lemma) . L et M b e a n R -mo dule. Then M is pr oje ctive if and only if ther e exists { m i } i ∈ I ⊆ M and { f i } i ∈ I ⊆ Hom R ( M , R ) , for some indexing set I , such that 1. for eve ry m ∈ M , f i ( m ) = 0 f o r al l b ut a ﬁnite subset of i ∈ I ; and 2. for eve ry m ∈ M , P i ∈ I f i ( m ) m i = m. Mor e o v er, I c a n b e chosen to b e a ﬁnite set if and only if M is ﬁni tely gener ate d. The collection { f i , m i } is called a dual b asis for M . Pro of. See [ 17 , Lemma I.1.3].  Chapter 1. Azuma ya Algebras 22 The follo wing prop o sition, from Villamay or, Zelinsky [ 69 , Prop. 1.1] (see also [ 17 , Prop. I I.2.1 ]) , s hows that an algebra whic h is separable and pro jectiv e is ﬁnitely gen- erated. This is a somewhat surprising result, as the requiremen t of b eing separable and pro jectiv e do es not immediately a pp ear to imply a ﬁnitely generated condition. The pro of of the prop osition uses the Dua l Basis Lemma. Prop osition 1.2.5. L et A b e a sep a r able R -algebr a which is pr oje c tive as an R - mo dule. T hen A is ﬁni tely gener a te d as an R -mo dule. Pro of. Since A is pro jectiv e as a n R -mo dule, A op is also pro j ectiv e a s an R - mo dule. Let { f i , a i } b e a dual basis for A op o v er R , where a i ∈ A op and f i ∈ Hom R ( A op , R ). Then for ev ery b ∈ A op , b = P i ∈ I f i ( b ) a i and f i ( b ) = 0 for all but ﬁnitely man y i ∈ I (using Lemma 1.2.4 ). Since A ⊗ R R ∼ = A , w e can identify A ⊗ R R with A and can consider id A ⊗ f i as a map from A e to A . This map is A -linear, and we claim that { id A ⊗ f i , 1 ⊗ a i } forms a dual basis for A e as a pro jectiv e left A -mo dule. Let a ⊗ b ∈ A e b e arbitrar y . Since f i ( b ) = 0 for all but a ﬁnite n umber of subscripts i , w e also ha ve ( id A ⊗ f i )( a ⊗ b ) = a ⊗ f i ( b ) = 0 for all but a ﬁnite n um b er of i . Then X i ∈ I (id A ⊗ f i )( a ⊗ b )(1 ⊗ a i ) = X i ∈ I a ⊗ f i ( b ) a i = a ⊗ b. Extended linearly , this holds fo r all u ∈ A e . So { id A ⊗ f i , 1 ⊗ a i } forms a dual basis for A e o v er A . Let a ∈ A op b e a ﬁxed a r bit r a ry elemen t. W e will sho w that a can b e written as an R -linear com bination of a ﬁnite subset o f A op , where this ﬁnite subset is indep enden t of a . Let e = P j x j ⊗ y j b e a separabilit y idempotent for A o v er R and deﬁne u = (1 ⊗ a ) e ∈ A e . W e ha v e u ∗ 1 = P j x j y j a = a and, fr o m Lemma 1.2.4 , Chapter 1. Azuma ya Algebras 23 u = P i (id A ⊗ f i ) ((1 ⊗ a ) e ) (1 ⊗ a i ). Then a = u ∗ 1 = X i   (id A ⊗ f i )((1 ⊗ a ) e ) ⊗ a i  ∗ 1  = X i  (id A ⊗ f i )((1 ⊗ a ) e )  · a i . (1.1) Using Prop osition 1.2.2 (5), (id A ⊗ f i )((1 ⊗ a ) e ) = (id A ⊗ f i )(( a ⊗ 1) e ) = a (id A ⊗ f i )( e ) = ( a ⊗ 1) ∗  (id A ⊗ f i )( e )  . Since { id A ⊗ f i , 1 ⊗ a i } forms a dual basis for A e , (id A ⊗ f i )( e ) = 0 for a ll but a ﬁnite subset of i ∈ I . So the set of subsc ripts i for whic h (id A ⊗ f i )  (1 ⊗ a ) e  is non-zero is con tained in the ﬁnite set of subscripts for whic h (id A ⊗ f i )( e ) is non-zero, whic h is indep enden t of a . Then since (id A ⊗ f i )  (1 ⊗ a ) e  is non-zero for only ﬁnitely many i ∈ I , the sum ( 1.1 ) ma y b e tak en ov er a ﬁnite set. Again writing e = P j x j ⊗ y j , ( 1.1 ) sa ys: a = X i,j ( x j ⊗ f i ( y j a )) · a i = X i,j x j f i ( y j a ) a i = X i,j f i ( y j a ) x j a i . So the ﬁnite set { x j a i } generates A op o v er R , and therefore generates A o v er R . This completes the pro of that A is ﬁnitely generated.  Chapter 1. Azuma ya Algebras 24 1.3 Other deﬁnitions of separabilit y In D eﬁnition 1.2.1 a b ov e, w e deﬁne separabilit y o ve r a c ommutativ e ring R . If R is a ﬁeld, w e also hav e the classical deﬁnition of separabilit y . F or a ﬁeld R , an R - algebra A is said to b e classi c al ly sep ar able if, for every ﬁeld extension L of R , the Jacobson radical of A ⊗ R L is zero, where the Jacobson r adical of A ⊗ R L is the in tersection of the maximal left ideals. The f o llo wing theorem sho ws the connection b et w een the t w o deﬁnitions of separabilit y when R is a ﬁeld. Theorem 1.3.1. L et R b e a ﬁeld and A b e an R -algebr a. T hen A is sep ar able over R if and only if A is classic a l ly sep ar ab le over R and the dimension of A as a ve ctor sp ac e ov er R i s ﬁ n ite. Pro of. See [ 17 , Thm. I I.2.5].  There is a further deﬁnition of separabilit y for ﬁelds. F or a ﬁeld R , an irreducible p olynomial f ( x ) ∈ R [ x ] is separable o ve r R if f has no repeat ed ro ots in an y splitting ﬁeld. An algebraic ﬁeld extension A of R is said to b e a sep ar able ﬁeld extension of R if, for ev ery a ∈ A , the minimal p olynomial of a o v er R is separable. The theorem b elo w sho ws that for a ﬁnite ﬁeld extension, this deﬁnition agr ees with the deﬁnition of classical separabilit y giv en ab ov e. The theorem also shows their connection with the deﬁnition of a separable algebra giv en in D eﬁnition 1.2.1 . Theorem 1.3.2. L et R b e a ﬁeld, and let A b e a ﬁnite ﬁeld extension of R . Then the fol lowing a r e e q uiva lent: 1. A is sep ar ab l e as an R -algebr a, 2. A is class i c al ly sep ar able over R , 3. A is a sep ar abl e ﬁeld extension of R . Pro of. (1) ⇔ (2): F ollows immediately from Theorem 1.3.1 . (2) ⇔ (3): See [ 71 , Lemma 9.2.8].  Chapter 1. Azuma ya Algebras 25 1.4 Azuma ya alge b ras Let R b e a commutativ e ring and A b e an R -algebra. There is a natural R -algebra homomorphism ψ A : A e → End R ( A ) deﬁned by ψ A ( a ⊗ b )( x ) = axb , extended linearly . If the con text is clear, w e will drop the subscript A . W e now are ready to deﬁne an Azuma y a a lgebra: this is the deﬁnition from [ 22 , p. 186] and [ 41 , p. 134]. Deﬁnition 1.4.1. An R -algebra A is called an Azumaya algebr a if the follo wing t wo conditions hold: 1. A is a faithfully pro jectiv e R -mo dule. 2. The map ψ A : A e → End R ( A ) deﬁned ab ov e is an isomorphism. Example 1.4.2. Any ﬁnite dimensional central simple alg ebra A ov er a ﬁeld F is an Az umay a alg ebra. A ce ntral simple algebra is free, so it is pro jectiv e and faithful, and w e kno w A ⊗ F A op ∼ = M n ( F ) ∼ = End F ( A ) (see [ 63 , Thm. 8.3.4]). W e will see some further examples of Azuma y a algebras on pages 28 and 3 2 . Prop osition 1.4.3. L et E 1 , E 2 , F 1 , F 2 b e R -mo dules. When on e of the or der e d p airs ( E 1 , E 2 ) , ( E 1 , F 1 ) , ( E 2 , F 2 ) c onsists of ﬁnitely gene r ate d pr oje c tive R -mo dules, the c anonic al homo m orphism Hom( E 1 , F 1 ) ⊗ Hom( E 2 , F 2 ) − → Hom( E 1 ⊗ E 2 , F 1 ⊗ F 2 ) is bije ctive. Pro of. See [ 7 , § I I.4 .4, Prop. 4].  W e observ e tha t if E 1 , . . . , E n , F 1 , . . . , F m are an y R - mo dules and φ : E 1 ⊕ · · · ⊕ E n − → F 1 ⊕ · · · ⊕ F m Chapter 1. Azuma ya Algebras 26 is an R -mo dule homomorphism, then φ can b e represen ted by a unique matrix      φ 11 · · · φ 1 n . . . . . . φ m 1 · · · φ mn      where φ ij ∈ Hom R ( E j , F i ). In pa rticular for R -mo dules M and N there are R -mo dule homomorphisms End R ( M ) i − → End R ( M ⊕ N ) j − → End R ( M ) f i 7− →   f 0 0 0     x y z w   j 7− → x Note that j ◦ i = id, so that i is injectiv e and j is surjectiv e. The prop osition b elo w sho ws how to construct some examples of Azuma y a alge- bras. The pro of is from F arb, Dennis [ 22 , Prop. 8.3]. Prop osition 1.4.4. I f P is a faithful ly pr oje ctive R -m o dule, then End R ( P ) is an Azumaya algebr a over R . Pro of. Using Prop osition 1.1.6 , P is a ﬁnitely generated pro jectiv e R - mo dule, so w e can c ho ose an R - mo dule Q with P ⊕ Q ∼ = R n for some n . Hence End R ( P ⊕ Q ) ∼ = End R ( R n ) ∼ = M n ( R ) ∼ = R n 2 as R - mo dules, so End R ( P ⊕ Q ) is a free module. W e sa w ab ov e that there are homomorphisms End R ( P ) i → End R ( P ⊕ Q ) j → End R ( P ) with j ◦ i = id . Then End R ( P ) is isomorphic to a direct summand of End R ( P ⊕ Q ) and therefore E nd R ( P ) is ﬁnitely generated and pro jectiv e as an R -mo dule. Supp ose r ∈ R annihilates End R ( P ). In particular r annihilates the iden tity map on P , and so r p = 0 f o r all p ∈ P . But P is a faithful mo dule, so r = 0, whic h sho ws that End R ( P ) is also a faithful R -mo dule. By Prop osition 1.1.6 , this sho ws that End R ( P ) is faithfully pro jectiv e. Chapter 1. Azuma ya Algebras 27 It remains to sho w the second condition. Consider the fo llo wing diagra m: End R ( P ) ⊗ End R ( P ) op i ′   ψ P / / End R (End R ( P )) i ′′   End R ( P ⊕ Q ) ⊗ (End R ( P ⊕ Q )) op j ′ O O ψ P ⊕ Q / / End R (End R ( P ⊕ Q )) j ′′ O O where ψ P and ψ P ⊕ Q are deﬁned as in Deﬁnition 1.4.1 , and the maps i ′ , j ′ , i ′′ and j ′′ come from the homomorphisms i a nd j on page 26 . F or a n elemen t f ⊗ g in End R ( P ) ⊗ End R ( P ) op , w e ha v e ψ P ⊕ Q ◦ i ′ ( f ⊗ g ) = i ′′ ◦ ψ P ( f ⊗ g ) : End R ( P ⊕ Q ) − → End R ( P ⊕ Q )   α 11 α 12 α 21 α 22   7− →   f ◦ α 11 ◦ g 0 0 0   and for f ⊗ g =   f 11 f 12 f 21 f 22   ⊗   g 11 g 12 g 21 g 22   ∈ End R ( P ⊕ Q ) ⊗ (End R ( P ⊕ Q )) op , w e ha v e ψ P ◦ j ′ ( f ⊗ g ) = j ′′ ◦ ψ P ⊕ Q ( f ⊗ g ) : End R ( P ) − → End R ( P ) α 7− → f 11 ◦ α ◦ g 11 . So the dia gram ab o v e comm utes and we can sho w that j ′′ ◦ i ′′ = id End(End( P )) and j ′ ◦ i ′ = id End( P ) ⊗ End( P ) op . Therefore to sho w that ψ P is an isomorphism it is su ﬃcien t to sho w that ψ P ⊕ Q is an isomorphism. Let { e 1 , . . . , e n } b e a basis for the free R -algebra P ⊕ Q ∼ = R n . L et E ij ∈ End R ( R n ) b e deﬁned by E ij ( e k ) = δ j k e i . W e can consider E ij as the n × n matrix with 1 in the i - j entry , and zeros elsewh ere. Then { E ij : 1 ≤ i, j ≤ n } is an R -mo dule basis f or End R ( R n ) and { E ij ⊗ E k l : 1 ≤ i, j, k , l ≤ n } is a n R -mo dule basis for End R ( R n ) ⊗ (End R ( R n )) op . By deﬁnition of ψ P ⊕ Q , w e ha v e ψ P ⊕ Q ( E ij ⊗ E k l )( E st ) = E ij ◦ E st ◦ E k l = δ j s δ tk E il . (1.2) Chapter 1. Azumay a Algebras 28 Using this, w e will show that ψ P ⊕ Q is an isomorphism. Let h : End R ( R n ) → End R ( R n ) b e an R -mo dule homomorphism. F or an arbitra r y basis elemen t E xy ∈ End R ( R n ), supp ose h ( E xy ) = P i,j r ( x,y ) ij E ij , where r ( x,y ) ij is an elemen t of R indexed b y i, j, x, y . Then deﬁne a map φ : End R (End R ( R n )) − → End R ( R n ) ⊗ (End R ( R n )) op h 7− → X x,y X i,j r ( x,y ) ij E ix ⊗ E y j . Then w e can sho w that φ is an R -algebra homomorphism inv erse to ψ P ⊕ Q , completing the pro of.  Example 1.4.5. F or any comm utativ e ring R , M n ( R ) is an Azuma y a algebra ov er R . This follo ws b y applying Prop osition 1.4.4 : since R n is fr ee and therefore faithfully pro jectiv e o ve r R , End R ( R n ) is an Azuma ya algebra ov er R , and w e kno w tha t M n ( R ) ∼ = End R ( R n ) as R -a lg ebras. In Prop osition 1.4.6 b elow w e show t ha t t he tensor pro duct of tw o R -Azuma y a algebras is again an R -Azumay a algebra. The pro of is from F arb, Dennis [ 22 , Prop. 8.4]. Prop osition 1.4.6. I f A and B ar e Azumaya alg e br as over R , then A ⊗ R B is an Azumaya algebr a over R . Pro of. Since A and B are ﬁnitely generated pro jectiv e R - mo dules (by Prop osi- tion 1.1.6 ), w e can choose R -mo dules A ′ and B ′ with A ⊕ A ′ ∼ = R n and B ⊕ B ′ ∼ = R m . Then ( A ⊕ A ′ ) ⊗ ( B ⊕ B ′ ) ∼ = R n ⊗ R m ∼ = R nm . So there is an R -mo dule Q with ( A ⊗ B ) ⊕ Q ∼ = R nm , pro ving A ⊗ B is ﬁnitely generated and pro j ectiv e. W e kno w that any pro jectiv e mo dule is ﬂat, so A is ﬂat o v er R . Since R → B ; r 7→ r. 1 B is injectiv e, it follows that f : A − → A ⊗ R R − → A ⊗ R B a 7− → a ⊗ 1 R 7− → a ⊗ 1 B Chapter 1. Azumay a Algebras 29 is injectiv e. Let r ∈ R b e suc h that r ∈ Ann( A ⊗ B ). Then r ( a ⊗ 1) = 0 for all a ∈ A , so 0 = r ( f ( a )) = f ( r a ), and since f is injectiv e t his implies tha t r a = 0 for all a ∈ A . Th us r ∈ Ann( A ) and so r = 0, as A is faithful, proving A ⊗ B is also faithful. So b y Prop osition 1.1.6 , A ⊗ B is fa ithfully pro jectiv e. Let ψ A ⊗ B b e the homomorphism deﬁned in the de ﬁnition of an Azuma y a algebra. Then the follow ing diagram is commutativ e: ( A ⊗ A op ) ⊗ ( B ⊗ B op ) θ   ψ A ⊗ ψ B / / End R ( A ) ⊗ End R ( B ) w   ( A ⊗ B ) ⊗ ( A ⊗ B ) op ψ A ⊗ B / / End R ( A ⊗ B ) Here ψ A and ψ B are isomorphisms since A and B are Azuma y a algebras, w is the isomorphism giv en b y Prop o sition 1.4.3 , and the isomorphism θ comes from the comm utativit y of the tensor pro duct and the fact that ( A ⊗ B ) op ∼ = A op ⊗ B op . This sho ws that ψ A ⊗ B is an isomorphism.  1.5 F urther c haracteris ati o ns of Azuma y a alge bras In t his section, unless otherwise stated, R denotes a comm utativ e ring and A is an R -algebra. The deﬁnition of a n Azuma y a alg ebra (Deﬁnition 1.4.1 ) has a num b er of equiv alen t reformulations whic h are show n in Theorem 1.5.3 b elo w. W e ﬁrstly require some additional deﬁnitions. W e sa y that an R -algebra A is c entr al o v er R if A is faithful as an R -mo dule and the cen tre of A coincides with the image of R in A . Th us an R -algebra A is cen tral if a nd o nly if the ring homomorphism f : R → Z ( A ) is b oth injectiv e and surjectiv e. A comm utative R -algebra S is said to b e a ﬁnitely pr esente d algebr a if S is isomorphic to the quotient ring R [ x 1 , . . . , x n ] /I of a p olynomial ring R [ x 1 , . . . , x n ] b y a ﬁnitely generated tw o-sided ideal I . A comm utative R -algebra S is called ´ etale if S is ﬂat, ﬁnitely presen ted and separable o v er R . Let R b e a ring, whic h is not necessarily commutativ e. Consider a cov aria nt Chapter 1. Azumay a Algebras 30 additiv e functor T fro m t he category of (left o r righ t) R -mo dules C to some category of mo dules D . The functor T induces a function T X,Y : Hom C ( X , Y ) − → Hom D ( T ( X ) , T ( Y )) for ev ery pair of ob jects X a nd Y in C . The functor T is said to b e • faithful if T X,Y is injectiv e; • ful l if T X,Y is surjectiv e; • ful ly faithful if T X,Y is bijectiv e; for each X and Y in C . An R -mo dule M is called a gener ator if the functor Hom R ( M , − ) is a faithful functor from the category of left R -mo dules to the category of ab elian groups. There is anot her deﬁnition of a generator mo dule in [ 17 , p. 5], whic h is sho wn to b e equiv a len t to the deﬁnition ab o v e (see Prop osition 1.5.1 ). F or a ring R (no t necessarily comm utative ) a nd an y R -mo dule M , consider the subset T R ( M ) o f R consisting of elemen ts of the f o rm P i f i ( m i ) w here the f i are from Hom R ( M , R ) and the m i are from M . Then T R ( M ) is a tw o- sided ideal of R , called the tr ac e ide al of M . Then [ 17 ] deﬁnes M to b e a generator mo dule if T R ( M ) = R . Prop osition 1.5.1. L et R b e a ( p ossibly non -c ommutative) ring. F or an y R -mo dule M , the f unc tor Hom R ( M , − ) is a fa i thful functor if and only if T R ( M ) = R . Pro of. See [ 43 , Thm. 18.8].  Let R b e a ring, whic h is not nec essarily commu tat iv e. An R -mo dule M is called a pr oje ctive gener ator ( or pr o gen er ator ) if M is ﬁnitely generated, pro jectiv e and a generator. Theorem 1.5.2. L et R b e a c om m utative ring. Then a n R -mo dule M is an R - pr o gener ator if and only if M is ﬁnitely gener ate d, pr oje ctive and faithful. Chapter 1. Azumay a Algebras 31 Pro of. See [ 17 , Cor. I.1.10].  Theorem 1.5.3. L et R b e a c ommutative ring and A b e an R -a l g e br a. The fol low i n g ar e e quivalent: 1. A is an Azumaya algebr a. 2. A is c entr al and sep ar able as a n R -mo dule. 3. A is c entr al over R and A is a gener ator as an A e -mo dule. 4. The f unctors A - M od - A − → M od - R, M 7− → M A M od - R − → A - M od - A, N 7− → N ⊗ A ar e inverse e q uiva l e nc es of c ate gorie s. F urther pr oje ctive mo dules c orr esp ond to pr oje ctive mo dules. 5. A is a ﬁnitely gen er ate d R -mo dule and A/ m A is a c entr al simple R/ m -algebr a for al l m ∈ Max( R ) . 6. Ther e is a faithful ly ﬂat ´ etale R -algebr a S and a faithful ly pr oje ctive S - m o dule P s uch that A ⊗ R S ∼ = End S ( P ) . If R is lo c al, S c an b e taken as ﬁnite ´ etale. Pro of. The pro of fo llo ws b y com bining v arious parts from [ 17 , Thm. I I.3.4], [ 4 , Thm. I I I.4.1] and [ 41 , Thm. I I I.5.1.1 ]. (1) ⇔ (2): See DeMey er, Ingraham [ 17 , Thm. I I.3.4]. (1) ⇔ (3): See Bass [ 4 , Thm. I I I.4.1]. (1) ⇔ (4) ⇔ (5) ⇔ (6): See Kn us [ 41 , Thm. I I I.5.1.1 ].  W e state another example o f an Azuma ya algebra, whic h follows from the theorem ab ov e. Chapter 1. Azumay a Algebras 32 Example 1.5.4. Let R b e a comm utative ring in which 2 is in v ertible. D eﬁne the quaternion algebra Q ov er R to b e the free R -mo dule with basis { 1 , i, j, k } and with m ultiplication satisfying i 2 = j 2 = k 2 = − 1 and ij = − j i = k . As for quaternion algebras ov er ﬁelds (see [ 52 , Lemma 1.6]), it f ollo ws that Q is a cen tra l R -algebra. Then [ 65 , Cor. 4 ] s ays that Q is separable o v er R . Their proof considers the following elemen t of Q : e = 1 4 (1 ⊗ 1 − i ⊗ i − j ⊗ j − k ⊗ k ) . It is ro utine to s how that e is a s eparability idem p oten t for Q , so Q is separable o v er R . It follow s from Theorem 1.5.3 (2) that Q is an Azuma y a a lgebra o v er R . Remark 1.5.5. If A is an Azuma y a algebra o v er R , then by Theorem 1.5.3 , A/ m A is a cen tra l simple R/ m-algebra for all m ∈ Max( R ). If A is free o ve r R , then [ A : R ] = [ R/ m ⊗ R A : R/ m] = [ A/ m A : R/ m] . Since w e kno w the dimension of a cen tral simple algebra is a square num b er (see [ 63 , Cor. 8.4.9]), the same is true for A . W e also rem ark that Azuma y a algebras are closely related to P olynomial Id entit y rings, thanks to t he Artin-Pro cesi Theorem (see [ 61 , § 1.8 ], [ 62 , § 6.1]) . While we do not consider p olynomial iden tity theory here, we mention a result of Braun in the follo wing theorem. In [ 9 ], Braun generalises the Artin-Pro cesi Theorem and, as a consequenc e of this generalisation, he giv es another c ha racterisation of Azuma ya algebras in [ 9 , Thm. 4.1], whic h w e state in the theorem b elo w. The pro of giv en b elo w is a direct pro of of this characterisation, whic h is due to Dic ks [ 19 ]. The notation used in the pro o f is deﬁned in Section 1.2 . Theorem 1.5.6. L et A b e a c en tr al R -algebr a. Then A i s an Azumaya algebr a over R if a nd only if ther e is some e ∈ A e such that e ∗ 1 = 1 and e ∗ A ⊆ R . Pro of. Using Theorem 1.5.3 , A is an Azuma y a algebra ov er R if and only if A is separable ov er R (since we assumed cen tral in the statemen t of the theorem). Assume Chapter 1. Azumay a Algebras 33 A is an Azumay a algebra o ve r R . By Theorem 1.2.2 (5), this implies there exists an idemp oten t e ∈ A e suc h t ha t e ∗ 1 = 1 and ( a ⊗ 1) e = (1 ⊗ a ) e fo r all a ∈ A . Let a ∈ A b e arbitrary . Then a ( e ∗ A ) = ( a ⊗ 1) ∗ ( e ∗ A ) = (( a ⊗ 1) e ) ∗ A = ((1 ⊗ a ) e ) ∗ A = (1 ⊗ a ) ∗ ( e ∗ A ) = ( e ∗ A ) a , proving e ∗ A ⊆ Z ( A ) = R . Con v ersely , supp ose there is some e ∈ A e with e ∗ 1 = 1 a nd e ∗ A ⊆ R . W e wan t to show that A is s eparable o ve r R ; that is, the re is an e ∈ A e suc h that e ∗ 1 = 1 and J e = 0 (using Theorem 1.2.2 ). W e will ﬁrst show that A e eA e = A e , where A e eA e is the ideal of A e generated b y e . Let I = { a ∈ A : a ⊗ 1 ∈ A e eA e } , whic h is a t w o-sided ideal of A . If I = A , then as 1 ∈ A , 1 ⊗ 1 ∈ A e eA e and so A e eA e = A e . Assume I 6 = A . Then there is a maximal ideal M of A suc h that I ⊆ M $ A . W e can giv e A e t w o left A e -mo dule structures as follow s: A e × A e − → A e ( u, a ⊗ b ) 7− → u ∗ 1 ( a ⊗ b ) = ( u ∗ a ) ⊗ b ( u, a ⊗ b ) 7− → u ∗ 2 ( a ⊗ b ) = a ⊗ ( u ∗ b ) It is routine to c hec k that these are w ell-deﬁned. Then e ∗ 2 ( a ⊗ b ) = a ⊗ ( e ∗ b ) = a ( e ∗ b ) ⊗ 1 a nd w e can show that A e ∗ 2 ( A e eA e ) ⊆ A e eA e . It f ollo ws that ev ery elemen t of e ∗ 2 ( A e eA e ) is of the form a ⊗ 1 for some a ∈ I ⊆ M . Let A = A/ M a nd R = R/ ( R ∩ M ). Then R ֒ → A and R ⊆ Z ( A ). Let A e = A ⊗ R A op . Then e ∗ 1 = 1 and e ∗ A ⊆ R . T o sho w R = Z ( A ), let x ∈ Z ( A ). Then x = e ∗ x , which shows x ∈ R since w e kno w e ∗ x ∈ R . Since M is a maximal ideal of A , A is simple, and therefore cen tral simple, ov er R . Also A op is simple, so A e is simple. Since e ∗ 1 = 1, e 6 = 0 and th us A e eA e = A e , as A e eA e is the t wo-side d ideal generated b y e . Then 1 ⊗ 1 = e ∗ 2 (1 ⊗ 1 ) ∈ e ∗ 2 A e eA e = e ∗ 2 A e eA e . But w e observ ed a b ov e that ev ery elemen t of e ∗ 2 A e eA e is o f t he form a ⊗ 1 for s ome a ∈ M , so ev ery elemen t of e ∗ 2 A e eA e is of the form a ⊗ 1 = a ⊗ 1 = 0, and therefore e ∗ 2 A e eA e = 0 . Since 1 ⊗ 1 ∈ e ∗ 2 A e eA e , this is a con tradiction, so our assumption that I 6 = A w as incorrect, pro ving A e eA e = A e . Chapter 1. Azumay a Algebras 34 It remains to prov e J e = 0. Let u ∈ A e . W e claim that A e eA e ∗ 2 u ⊆ ( u ∗ 1 A e ) A e . Let v , w ∈ A e and let u = P i a i ⊗ b i . Then ( v ew ) ∗ 2 ( X i a i ⊗ b i ) = X i ( a i ⊗ ( v ew ∗ b i )) = X i  a i ⊗  v ∗  e ∗ ( w ∗ b i )    But e ∗ ( w ∗ b i ) ∈ R as e ∗ A ⊆ R , so v ∗  e ∗ ( w ∗ b i )  = ( v ∗ 1)( ew ∗ b i ). Then letting ew = P j c j ⊗ d j , w e hav e ( v ew ) ∗ 2 ( X i a i ⊗ b i ) = X i a i ( ew ∗ b i ) ⊗ ( v ∗ 1 ) = X i X j ( a i c j b i d j ) ⊗ ( v ∗ 1) = X j   X i a i ⊗ b i  ∗ 1  c j ⊗ ( v ∗ 1)   ( d j ⊗ 1) , pro ving the claim. Since ∗ deﬁnes a left mo dule a ction on A and since e ∗ A ⊆ R , it fo llo ws that J e ∗ A = J ∗ ( e ∗ A ) ⊆ J ∗ R . Then it is easy to see that J ∗ R = 0, and so J e ∗ A = 0 . But J e ⊆ A e ∗ 2 ( J e ) = A e eA e ∗ 2 ( J e ) ⊆ ( J e ∗ 1 A e ) A e = 0, prov ing J e = 0 as required.  W e recall that for t w o r ings A and A ′ , an anti-homomorphism of A in to A ′ is a map σ : A → A ′ satisfying the following conditions: 1. σ ( a + b ) = σ ( a ) + σ ( b ) for all a, b ∈ A ; 2. σ (1 A ) = 1 A ′ ; 3. σ ( ab ) = σ ( b ) σ ( a ) for all a, b ∈ A . If A = A ′ and σ : A → A is a bijectiv e anti-homomorphism, then σ is called an an ti- automorphism. An involution is an an ti-a ut o morphism satisfying the additional condition: Chapter 1. Azumay a Algebras 35 4. σ 2 ( a ) = a for all a ∈ A . If A is a cen tral R -algebra, then an in v olution σ of A is said to b e of the ﬁrst ki n d if the restriction of σ to R is the iden tity map. W e note that an in v olution of the ﬁrst kind is an R -linear inv olution; that is, the map σ : A → A is an R -linear map. F or a cen tra l R - algebra A admitting a n in v olution of the ﬁrst kind, Braun [ 10 , Thm. 5] giv es a further c haracterisation of when A is an Azuma ya a lgebra, whic h is stated in Theorem 1.5.7 . The theorem sho ws that, in this setting, the condition that A is a ﬁnitely generated pro jectiv e R -mo dule is not required. Theorem 1.5.7. L et A b e a c entr al R -algebr a ad m itting a n involution of the ﬁrst kind. Then A is a n Azumaya algebr a if an d only if ψ A : A ⊗ R A op → End R ( A ) is an R -line ar iso m orphism. Pro of. See Braun [ 10 , Thm. 5].  This theorem has b een further generalised by Ro w en [ 60 , Cor. 1.7]. F or a cen tral R -algebra A admitting a n inv olution of the ﬁrst kind, the result of Ro w en prov es that for A to b e an Azuma y a algebra, it is suﬃcien t to assume that ψ A is an epimorphism. 1.6 The de v e lopmen t of the the o ry of Azuma y a algebras The ab o ve results show some o f the most general reformulations of the deﬁnition o f an Azuma y a alg ebra t o date. W e will no w consider the historical dev elopmen t of these concepts, b eginning with the 1951 pap er of Azuma y a [ 3 ]. In [ 3 ], Azuma y a in tro duced the term “prop er maximally cen tral algebra” (p. 128). An R -algebra A whic h is f ree and ﬁnitely generated as a mo dule o ve r R is deﬁned to b e pr op er m aximal ly c entr al o v er R if A ⊗ A op coincides with End R ( A ). It is kno wn that a free R -mo dule is b oth faithful and pro jectiv e as a mo dule o v er R , so this deﬁnition implies that A is f a ithfully pro j ectiv e as an R -mo dule, and A ⊗ A op ∼ = Chapter 1. Azumay a Algebras 36 End R ( A ). This show s that the deﬁnition of a prop er maximally cen tral algebra is equiv alen t to Deﬁnition 1.4.1 under the additional assumption that A is free. Assuming that R is a No etherian ring, Auslander and Goldman pro ve Theo- rem 1.2.3 (see [ 2 ], Cor. 4.5 and Thm. 4.7). F urther, Endo and W a tanab e [ 2 1 , Prop. 1.1 ] g eneralise this result by remo ving the No etherian condition on R , as stated ab ov e in Theorem 1.2.3 . The equiv alence of statemen ts (1) and (2) of Theorem 1.5.3 was prov en b y Auslan- der and Goldman [ 2 , Thm. 2 .1 ] under the assumption that A is a c entral algebra o v er R , whic h w e can see in Theorem 1.5.3 is not required. In Kn us [ 41 , Thm. I I I.5.1 .1(2)], Theorem 1.5.3 (2) has the condition that A is ﬁnitely generated, whic h isn’t required for the equiv alence of the statemen ts. How ev er, w e note that giv en the equiv alence of (1) and (2), the fact that A is ﬁnitely generated follows from Prop osition 1.2.5 . P art (3) of Theorem 1.5.3 generalises a result of DeMey er, Ingraham [ 17 , Thm. I I.3.4(2)]. Their result sa ys that A is central ov er R and A is a progenerator o v er A e if and only if A is a n Azuma y a algebra ov er R . But w e can see that Hom A e ( A, − ) b eing an exact functor is sup erﬂuous. Azuma y a [ 3 , Thm. 1 5 ] pro v es the equiv a lence of Theorem 1.5.3 parts (1) and (5) with the extra condition that A is free o v er R . F urther, with the assumption tha t A is pro jective , Bass [ 4 , Thm. I I I.4.1] prov es this equiv a lence. Theorem 1.5.3 sho ws that neither of these extra conditions on A a re r equired for the equiv alence of these t w o statemen ts. Bass [ 4 , Thm. I I I.4.1] show s that the deﬁnition of an Azuma ya algebra is equiv alent to the following: There e xists an R -algebra S and a faithfully pro jectiv e R -mo dule P suc h that A ⊗ R S ∼ = End R ( P ). Comparing this with Theorem 1.5.3 (6), the result of Knus [ 41 , Thm. I I I.5.1.1(5)] reﬁnes this result b y giving that S is a faithf ully ﬂat ´ etale R - algebra. W e pro v e our main theorem of Chapter 3 , Theorem 3.1.5 , for Azumay a algebras whic h a r e free o v er their cen tres. So for an Azuma y a algebra A as orig inally deﬁned b y Azuma ya ( with A f r ee o ve r its cen t re R ), our Theorem 3.1.5 cov ers its K -theory . Chapter 2 Algebraic K -Theory Algebraic K -t heory deﬁnes a sequence of functors K i ( R ), for i ≥ 0, from the catego r y of rings to the category of ab elian groups. The low er K -groups K 0 , K 1 and K 2 w ere dev elop ed in the 1950s and 60s b y G rothendiec k, Bass and Milnor resp ectiv ely . After m uc h uncertain t y , the “correct” deﬁnition of the higher K - groups w as give n b y Quillen in 1974 . F or an in tro duction to the lo w er K -groups, see Magurn [ 48 ] or Silv ester [ 64 ], and for an in tro duction to higher algebraic K -theory , see Ro sen b erg [ 58 ] or W eib el [ 70 ]. W e b egin this c hapter b y recalling the deﬁnitions of K 0 , K 1 and K 2 , and by observing some pro p erties a nd examples of these lo we r K -groups. W e then s p ecialise to the low er K - groups of cen tral simple alg ebras. Lastly , w e lo ok at the hig her K - groups, and note some of their prop erties. 2.1 Lo w er K -groups K 0 Let R b e a ring . Let P ro j( R ) denote the mo no id of isomorphism classes of ﬁnitely generated pro jectiv e R -mo dules, with direct sum a s the binary o p eration and the zero mo dule as t he iden tity elemen t. The n K 0 ( R ) is deﬁned to b e the free a b elian 37 Chapter 2. Algebraic K - Theory 38 group based on P ro j( R ) mo dulo the subgroup generated by elemen ts of the fo r m [ P ] + [ Q ] − [ P ⊕ Q ], for P , Q ∈ P r( R ). Alternat ively , the group K 0 ( R ) c an b e deﬁned as the group completion of the monoid P ro j( R ) , whic h is sho wn in [ 58 , Thm. 1.1.3] to b e equiv alent to the deﬁnition giv en here. By [ 58 , p. 5, Remarks], t he group completion construction fo r ms a functor from the category of ab elian semigroups to the category of ab elian groups. Theorem 2.1.1. F or rings R and S , let T b e a n additive functor fr om the c ate gory of R -mo dules to the c ate gory of S -mo d ules , with T ( R ) ∈ P r( S ) . Then T r estricts to an ex a ct functor fr om P r( R ) to P r( S ) , which induc es a gr oup homomo rphism f : K 0 ( R ) → K 0 ( S ) with f ([ P ]) = [ T ( P )] for e ac h P ∈ P r ( R ) . Pro of. (See [ 48 , Prop. 6.3].) If P ∈ P r( R ), then P ⊕ Q ∼ = R n for some R -mo dule Q , a nd we hav e T ( P ) ⊕ T ( Q ) ∼ = T ( R ) n . Since T ( R ) n ∈ P r( S ), it follo ws that the functor T tak es P r( R ) to P r( S ). All short exact sequences in P r( R ) are split, so the restriction of T to P r( R ) → P r( S ) is a n exact functor. Since functors preserv e isomorphisms, T induces a monoid homomorphism P ro j( R ) − → P ro j( S );  P  7− →  T ( P )  . Since the group completion is functorial, there is a group homomorphism K 0 ( R ) → K 0 ( S );  P  7→  T ( P )  .  F or ring s R and S , if φ : R → S is a ring homomo r phism, then S is a right R -mo dule via φ . There is an additiv e functor S ⊗ R − : R - M od − → S - M o d whic h, b y Theorem 2.1.1 , induces a gr o up homomorphism f : K 0 ( R ) → K 0 ( S ); f ([ P ]) = [ S ⊗ R P ] for eac h P ∈ P r ( R ). Theorem 2.1.2. K 0 is a functor fr om the c ate gory of ri n gs to the c a te gory of ab e lian gr oups. Chapter 2. Algebraic K - Theory 39 Pro of. See [ 48 , Thm. 6.22] for the details.  Theorem 2.1.3. F or rings R and S , ther e is a gr oup isomorphism K 0 ( R × S ) ∼ = K 0 ( R ) × K 0 ( S ) . Pro of. See [ 48 , Thm. 6.6].  Recall that rings R and S are Morita equiv alen t, if R - M od and S - M od are equiv- alen t as categories; that is, there are functor s R - M od T / / S - M od U o o with natural equiv alences T ◦ U ∼ = id S and U ◦ T ∼ = id R . It follows from [ 49 , Prop. I I.10.2 ] that T and U are additiv e functors. In particular, M n ( R ) is Morita equiv alen t to R since the functors R n ⊗ M n ( R ) − : M n ( R ) - M od − → R - M od and R n ⊗ R − : R - M od − → M n ( R ) - M od form m utually inv erse equiv alences of categories. Theorem 2.1.4. F or rings R and S , if R and S ar e Morita e quivalent then K 0 ( R ) ∼ = K 0 ( S ) . In p articular, K 0 ( R ) ∼ = K 0 ( M n ( R )) . Pro of. (See [ 70 , Cor. I I.2.7.1].) This follows from Theorem 2.1 .1 since the functors R - M od T / / S - M od U o o induce an equiv a lence b et w een t he categories P r( R ) and P r( S ), which induces a group isomorphism on the lev el of K 0 .  Examples 2.1.5. 1. If R is a ﬁeld or a division ring, then K 0 ( R ) ∼ = Z . A ﬁnitely generated pro j ective module ov er R is free, with a uniquely deﬁned dimension. Chapter 2. Algebraic K - Theory 40 Since a n y t wo ﬁnite dimensional free mo dules with the same dimension are isomorphic, w e hav e P r o j( R ) ∼ = N , so K 0 ( R ) ∼ = Z . 2. Recall that a ring R is a semi-simple ring if it is semi-simple as a left mo dule o v er itself; tha t is, R is a direct sum of simple R - submo dules. By the Artin- W edderburn theorem, R is isomorphic to M n 1 ( D 1 ) × · · · × M n r ( D r ), w here e ach D i is a division ring and eac h n i is a p ositiv e in teger. Then using Theorems 2.1.3 and 2.1.4 , K 0 ( R ) ∼ = M K 0  M n i ( D i )  ∼ = M K 0 ( D i ) ∼ = Z r . 3. If R is a principal ideal domain (a comm utativ e integral domain in whic h ev ery ideal can b e generated b y a single elemen t), or if R is a lo cal ring, then K 0 ( R ) ∼ = Z . F o r a pro of of these, see [ 58 ], Thm. 1.3 .1 and Thm. 1.3.11 resp ectiv ely . K 1 Let G L n ( R ), the general linear group of R , denote the g roup o f in v ertible n × n matrices with en t r ies in R and mat r ix multiplication as the binary op eration. Then GL n ( R ) is em b edded in to GL n +1 ( R ) via GL n ( R ) − → GL n +1 ( R ) M n 7− →   M n 0 0 1   . The union of the r esulting sequence GL 1 ( R ) ⊂ GL 2 ( R ) ⊂ · · · ⊂ G L n ( R ) ⊂ · · · is called the inﬁnite general linear group GL( R ) = S ∞ n =1 GL n ( R ). Then K 1 ( R ) is deﬁned to b e the ab elianisation o f GL ( R ); that is, K 1 ( R ) = GL( R ) / [GL ( R ) , GL( R )] . Chapter 2. Algebraic K - Theory 41 By [ 48 , Prop. 9.3], the ab elianisation construction forms a functor from the c atego ry of groups to the category of ab elian groups. The elemen tary matrix e ij ( r ), for r ∈ R a nd i 6 = j , 1 ≤ i, j ≤ n , is deﬁned to b e the matrix in G L n ( R ) whic h has 1’s on the diagonal, r in the i - j entry and zeros elsewhere . Then E n ( R ) denotes the subgroup of GL n ( R ) generated by all elemen tary matrices e ij ( r ) with 1 ≤ i, j ≤ n . W e note that the in ve rse of the matrix e ij ( r ) is e ij ( − r ) and the elemen t ary matrices satisfy the follo wing relations: e ij ( r ) e ij ( s ) = e ij ( r + s ) [ e ij ( r ) , e k l ( s )] =          1 if j 6 = k , i 6 = l e il ( r s ) if j = k , i 6 = l e k j ( − sr ) if j 6 = k , i = l . Then E n ( R ) em b eds in E n +1 ( R ), and E ( R ) is the inﬁnite union of the E n ( R ). The Whitehead Lemma (see [ 48 , Lemma 9.7]) states that [GL( R ) , GL( R )] = E ( R ), so it follo ws that K 1 ( R ) = GL( R ) /E ( R ) . Theorem 2.1.6. F or rings R and S a nd a ring ho momorphism φ : R → S , ther e is an induc e d gr oup homomorphism K 1 ( R ) → K 1 ( S ) takin g ( a ij ) E ( R ) to ( φ ( a ij )) E ( S ) . Then K 1 is a functor fr om the c ate gory of rings to the c ate gory of ab elian gr oups. Pro of. A ring homomor phism φ : R → S deﬁnes a group homomorphism GL( R ) → GL( S ); ( a ij ) 7→ ( φ ( a ij )). Then there is an induced g r o up homomorphism K 1 ( R ) → K 1 ( S ) taking ( a ij ) E ( R ) to ( φ ( a ij )) E ( S ). See [ 48 , Prop. 9.9] for the details.  Theorem 2.1.7. F o r a ring R and a p ositive inte ger n , K 1 ( M n ( R )) ∼ = K 1 ( R ) . Pro of. (See the pro of of [ 58 , Thm. 1.2.4].) Since M k ( M n ( R )) ∼ = M k n ( R ), w e hav e a Chapter 2. Algebraic K - Theory 42 comm utativ e dia gram: GL k ( M n ( R )) ∼ = / /   GL k n ( R )   GL k +1 ( M n ( R )) ∼ = / / GL ( k +1) n ( R ) . It f ollo ws that G L( M n ( R )) ∼ = GL( R ), whic h induces the required isomorphism on the lev el of K 1 .  Examples 2.1.8. 1. If R is a ﬁeld then K 1 ( R ) ∼ = R ∗ . See [ 70 , Eg. I I I.1.1.2] fo r the details. This follows since, f o r a comm utative ring R , the determinan t det : GL( R ) → R ∗ induces a split surjectiv e group ho mo mo r phism det : K 1 ( R ) → R ∗ ; ( a ij ) E ( R ) 7→ det( a ij ), with righ t inv erse giv en b y R ∗ → K 1 ( R ); a 7→ aE ( R ). When R is a ﬁeld , the k ernel of the map det is trivial, so K 1 ( R ) ∼ = R ∗ . 2. If R is a division ring, then K 1 ( R ) ∼ = R ∗ / [ R ∗ , R ∗ ]. See [ 70 , Eg. I I I.1.3.5] for the details. This follo ws f r om the Dieudonn ´ e determinan t (see [ 64 , p. 124]). The Dieudonn´ e determinan t deﬁnes a surjectiv e gro up homo mo r phism Det : GL n ( R ) → R ∗ / [ R ∗ , R ∗ ] with k ernel E n ( R ), suc h that Det      x 1 0 . . . 0 x n      = Y i x i [ R ∗ , R ∗ ] and the follo wing diagram commutes : GL n ( R ) / / Det % % K K K K K K K K K GL n +1 ( R ) Det x x r r r r r r r r r r R ∗ /R ′ It follow s that for a division ring R , K 1 ( R ) ∼ = R ∗ / [ R ∗ , R ∗ ]. If R is a ﬁeld, then the Dieudonn ´ e determinan t coincides with the usual determinan t. Chapter 2. Algebraic K - Theory 43 3. If R is a semi-lo cal ring (sub ject to some conditions, detailed b elow ), then K 1 ( R ) ∼ = R ∗ / [ R ∗ , R ∗ ]. See [ 68 , Thm. 2 ] for the details. Recall that R is a semi-lo cal ring if R/ rad( R ) is se mi-simple, where rad( R ) denotes t he Jacobson radical of R . By the Artin-W edderburn Theorem, R / rad( R ) is isomorphic to a ﬁnite pro duct o f matrix rings o ve r division rings. W e assume tha t none of these matrix ring s are isomorphic to M 2 ( Z / 2 Z ) a nd that no more than one of the matrix rings has order 2. The required result follow s since there is a Whitehead determinan t whic h giv es a surjectiv e group homomorphism R ∗ → K 1 ( R ). V aserstein [ 68 ] sho ws that the ke rnel o f this map is [ R ∗ , R ∗ ], giving the required isomorphism. K 2 F or n ≥ 3, the Stein b erg group St n ( R ) of R is the group deﬁned by generators x ij ( r ), with i , j a pair of dis tinct in tegers b et w een 1 and n and r ∈ R , sub ject to the follo wing relations which a re called the Stein b erg relations: x ij ( r ) x ij ( s ) = x ij ( r + s ) [ x ij ( r ) , x k l ( s )] =          1 if j 6 = k , i 6 = l x il ( r s ) if j = k , i 6 = l x k j ( − sr ) if j 6 = k , i = l . W e o bserv ed on pag e 41 that the elemen tary matrices s atisfy the Stein b erg relations. So there is a surjectiv e homomo r phism St n ( R ) → E n ( R ) sending x ij ( r ) to e ij ( r ). As the Stein b erg relations for n + 1 include the Steinberg relations for n , there are natural maps St n ( R ) → St n +1 ( R ). W e write St( R ) for the direct limit lim − → St n ( R ). Then there is a canonical surjectiv e map St( R ) → E ( R ), and K 2 ( R ) is deﬁned to b e the k ernel of this map. Theorem 2.1.9. K 2 is a functor fr om the c ate gory of ri n gs to the c a te gory of ab e lian gr oups. Chapter 2. Algebraic K - Theory 44 Pro of. See [ 48 , Prop. 12.6].  Theorem 2.1.10. F or rings R and S , if R and S ar e Morita e quivalent then K 2 ( R ) ∼ = K 2 ( S ) . In p articular, sinc e M n ( R ) is Morita e quivalent to R , K 2 ( R ) ∼ = K 2 ( M n ( R )) . Pro of. See [ 70 , Cor. I I I.5.6.1].  Example 2.1.11. 1. If R is a ﬁnite ﬁeld, then K 2 ( R ) = 0. See [ 58 , Cor. 4 .3.13]. 2. If R is a ﬁeld, then Matsumoto’s Theorem sa ys tha t K 2 ( R ) = R ∗ ⊗ Z R ∗ / h a ⊗ (1 − a ) : a 6 = 0 , 1 i . See [ 58 , Thm. 4.3.15]. 3. If R is a division ring, let U R denote the gro up generated by c ( x, y ), x, y ∈ R ∗ , sub ject to the relations: (U0) c ( x, 1 − x ) = 1 ( x 6 = 1 , 0), (U1) c ( xy , z ) = c ( xy x − 1 , xz x − 1 ) c ( x, z ) (U2) c ( x, y z ) c ( y , z x ) c ( z , xy ) = 1 . Then there is an exact sequence 0 − → K 2 ( R ) − → U R − → [ R ∗ , R ∗ ] − → 0 where U R → [ R ∗ , R ∗ ]; c ( x, y ) 7→ [ x, y ]. See Rehmann [ 5 5 , Cor. 2 , p. 10 1 ]. 2.2 Lo w er K -groups of ce ntral simple alge bras W e now sp ecialise to central simple algebras. In this section, all cen tral simple algebras are assumed to b e ﬁnite dimensional. Let F b e a ﬁeld and let A b e a Chapter 2. Algebraic K - Theory 45 cen tral simple a lg ebra ov er F . Clearly there is a ring homomorphism F → A . Since K 0 is a functor from the category o f rings to t he category of ab elian groups (see Theorem 2.1.2 ), w e ha ve a n exact seque nce 0 − → ZK 0 ( A ) − → K 0 ( F ) − → K 0 ( A ) − → CK 0 ( A ) − → 0 (2.1) where ZK 0 ( A ) and CK 0 ( A ) are the k ernel and cok ernel of the map K 0 ( F ) → K 0 ( A ), resp ectiv ely . In this section, using the deﬁnition of K 0 , w e will s how what seque nce ( 2.1 ) lo oks lik e. Using this, w e will observ e that CK 0 ( A ) and ZK 0 ( A ) are torsion ab elian g roups, and that CK 0 and ZK 0 are functors which do not resp ect Morita equiv alence, from the category o f cen tra l simple algebras o ve r F to the category of ab elian groups. W e ha v e a similar exact sequence for K 1 , and w e sho w that the same results hold. In Chapter 3 , w e will generalise this result to cov er Azumay a a lgebras whic h are free o v er their cen tres, and to co ve r all K i groups fo r i ≥ 0 . This is the k ey to pro ving that K i ( A ) ⊗ Z [1 /n ] ∼ = K i ( R ) ⊗ Z [1 /n ] for an Azuma ya algebra A free o ve r its cen tr e R of dimension n (see Theorem 3.1.5 ). K 0 Let A b e a cen tral simple algebra ov er a ﬁeld F . W edderburn’s theorem sa ys that A is isomorphic to M n ( D ) for a unique division algebra D and a unique p ositiv e in teger n . Since K 0 is a functor from t he category of rings to the category of abelian groups, we ha v e K 0 ( A ) ∼ = K 0  M n ( D )  , and similarly for CK 0 and Z K 0 . W riting M n ( D ) instead of A , sequence ( 2.1 ) can b e written as 0 − → ZK 0  M n ( D )  − → K 0 ( F ) − → K 0  M n ( D )  − → CK 0  M n ( D )  − → 0 . Chapter 2. Algebraic K - Theory 46 F rom the ring homomorphism F → M n ( D ) and since M n ( D ) is Morita equiv alen t to D , there are induced functors P r( F ) − → P r( M n ( D )) − → P r( D ) P ∼ = F k 7− → M n ( D ) k 7− → D n ⊗ M n ( D ) M n ( D ) k ∼ = D k n , where ev ery ﬁnitely generated pro jectiv e mo dule P ov er F is free. By Theorem 2.1.1 , these induce group homomorphisms K 0 ( F ) γ − → K 0  M n ( D )  δ − → K 0 ( D )  F k  7− →  M n ( D ) k  7− →  D k n  where, for [ X ] ∈ K 0  M n ( D )  , δ ([ X ]) is deﬁned to b e  D n ⊗ M n ( D ) X  . Since F is a ﬁeld and D is a division ring, f r o m Example 2.1.5 (1), K 0 ( F ) ∼ = Z ; [ F k ] 7→ k and K 0 ( D ) ∼ = Z ; [ D k ] 7→ k . W e hav e a commu tat ive diagram: 0 / / ZK 0  M n ( D )    / / K 0 ( F ) id   γ / / K 0  M n ( D )  δ   / / CK 0  M n ( D )    / / 0 0 / / k er( δ ◦ γ ) / /   K 0 ( F ) δ ◦ γ / / ∼ =   K 0 ( D ) / / ∼ =   cok er ( δ ◦ γ )   / / 0 0 / / 0 / / Z η n / / Z / / Z n / / 0 where the map η n : Z → Z is deﬁned by η n ( k ) = k n . W e know that M n ( D ) is Morita equiv alent to D , so δ is an isomorphism and thus all t he ve rtical maps in the ab ov e comm utative dia gram are isomorphisms. So the exact sequen ce ( 2 .1 ) can b e written as: 0 − → Z η n − → Z − → Z /n Z − → 0 (2.2) since ZK 0  M n ( D )  ∼ = k er( η n ) = 0 and CK 0  M n ( D )  ∼ = cok er ( η n ) = Z n . A g r oup G is said to b e n -torsion if x n = e f or all x ∈ G . W e note tha t clearly b o th ZK 0  M n ( D )  and CK 0  M n ( D )  are n -torsion groups. Chapter 2. Algebraic K - Theory 47 Let F b e a ﬁxed ﬁeld. W e will observ e that CK 0 and ZK 0 form functors from the category of cen tral simple alg ebras o v er F (with F - a lgebra homomorphisms) to the category of ab elian groups. F or any central simple algebra A o v er F , CK 0 ( A ) is deﬁned t o b e the cok ernel of the map K 0 ( F ) → K 0 ( A ); [ P ] 7→ [ A ⊗ F P ], and ZK 0 is its k ernel. They are clearly b oth ab elian gr o ups since K 0 ( F ) and K 0 ( A ) are ab elian g roups. F or an y t w o cen tra l simple algebras A and B ov er F with φ : A → B an F - algebra homomorphism, then φ restricted to F g iv es the iden tity map on F . There is a comm utativ e diagram: 0 / / ZK 0 ( A )   / / K 0 ( F ) id   / / K 0 ( A ) φ   / / CK 0 ( A )   / / 0 0 / / ZK 0 ( B ) / / K 0 ( F ) / / K 0 ( B ) / / CK 0 ( B ) / / 0 where φ : K 0 ( A ) → K 0 ( B ) is deﬁned by φ ([ P ]) = [ B ⊗ A P ] for eac h [ P ] ∈ K 0 ( A ). One can easily c hec k tha t CK 0 and ZK 0 form the required functors. W e also o bserv e tha t CK 0 do es no t resp ect Morita inv ariance. F o r a division algebra D o v er the ﬁeld F , we kno w that D is Morita equiv alent to M n ( D ). F or D , the exact sequen ce ( 2.1 ) can b e written as 0 − → 0 − → Z id − → Z − → 0 − → 0 since the homomorphism K 0 ( F ) → K 0 ( D ) maps [ F k ] to [ D ⊗ F F k ] = [ D k ]. F rom sequence ( 2.2 ) w e sa w that CK 0 ( M n ( D )) ∼ = Z n , whic h is clearly not isomorphic to CK 0 ( D ) = 0. K 1 Let A b e a cen tral simple algebra o ve r a ﬁeld F , suc h that A is isomorphic t o M r ( D ) for a uniq ue division algebra D and a unique po sitiv e integer r . Since K 1 is a func tor from the catego ry of rings to t he category of ab elian groups (see Theorem 2.1.6 ), Chapter 2. Algebraic K - Theory 48 w e will write M r ( D ) instead of A , since K 1 ( A ) ∼ = K 1  M r ( D )  . W e hav e an exact sequence 1 − → ZK 1  M r ( D )  − → K 1 ( F ) − → K 1  M r ( D )  − → CK 1  M r ( D )  − → 1 , (2.3) where ZK 1  M r ( D )  and CK 1  M r ( D )  are the ke rnel and cok ernel of the map K 1 ( F ) → K 1  M r ( D )  resp ectiv ely . The ring homomorphism F → M r ( D ); f 7→ f · I r induces a map GL( F ) − → GL( M r ( D )) ( a ij ) 7− → ( a ij I r ) . Using Theorems 2.1.6 and 2.1.7 , there are induced group homo mo r phisms K 1 ( F ) γ − → K 1  M r ( D )  δ − → K 1 ( D ) ( a ij ) E ( F ) 7− → ( a ij I r ) E ( M r ( D )) ( d ij ) E ( M r ( D )) 7− → ( d ij ) E ( D ) . So w e ha ve a comm utat ive diagram: 1 / / ZK 1  M r ( D )    / / K 1 ( F ) id   γ / / K 1  M r ( D )  δ   / / CK 1  M r ( D )    / / 1 1 / / k er( δ ◦ γ ) / /   K 1 ( F ) δ ◦ γ / / det   K 1 ( D ) / / Det   cok er ( δ ◦ γ )   / / 1 1 / / D ′ ∩ F ∗ r / / F ∗ β / / D ∗ /D ′ / / D ∗ /F ∗ r D ′ / / 1 . The maps det and Det come f rom Examples 2.1.8 and t he map β : F ∗ → D ∗ /D ′ is deﬁned b y β ( f ) = Det ◦ ( δ ◦ γ ) ◦ det − 1 ( f ) = f r D ′ , where D ′ = [ D ∗ , D ∗ ]. W e sa w in Examples 2.1.8 (1),(2) that K 1 ( F ) ∼ = F ∗ and K 1 ( D ) ∼ = D ∗ /D ′ . So all the v ertical maps in the ab ov e diagram are isomorphisms and the exact sequence Chapter 2. Algebraic K - Theory 49 ( 2.3 ) can b e written as 1 − → D ′ ∩ F ∗ r − → F ∗ − → D ∗ /D ′ − → D ∗ /F ∗ r D ′ − → 1 . (2.4) W e will sho w that the groups ZK 1  M r ( D )  ∼ = D ′ ∩ F ∗ r and CK 1  M r ( D )  ∼ = D ∗ /F ∗ r D ′ are b oth nr - torsion groups for n = ind( D ), where ind( D ) , the index of D o v er F , is deﬁned to b e t he square ro ot o f the dimension of D o ve r F . Note that b y [ 63 , Cor. 8.4.9], the dimension of D o v er F is a square n um b er. Lemma 2.2.1. L et D b e a division algebr a wi th c entr e F of index n . Then for any a ∈ D , a n = Nrd D ( a ) d a wher e d a ∈ D ′ . Pro of. Let a ∈ D b e ar bitr ary and let f a ( x ) b e the minimal p olynomial of a of degree m . By W edderburn’s F actorisation Theorem (see [ 42 , Thm. 16 .9]), f a ( x ) = ( x − d 1 ad 1 − 1 ) · · · ( x − d m ad m − 1 ) where d i ∈ D , and by [ 56 , p. 124, Ex. 1], w e ha v e f a ( x ) n/m = x n − T rd D ( a ) x n − 1 + · · · + ( − 1) n Nrd D ( a ). Then Nrd D ( a ) = ( d 1 ad − 1 1 · · · d m ad − 1 m ) ( n/m ) = ([ d 1 , a ] a [ d 2 , a ] a · · · a [ d m , a ] a ) ( n/m ) = a n d ′ a where d ′ a ∈ D ′ . So a n = Nrd D ( a ) d a for some d a ∈ D ′ .  F or a r ∈ Z K 1  M r ( D )  ∼ = D ′ ∩ F ∗ r , w e hav e a r ∈ F ∗ r ⊆ F ∗ , so ( a r ) n = Nrd D ( a r ) where n = ind( D ). Since a r ∈ D ′ , Nrd D ( a r ) = 1, proving ZK 1  M r ( D )  is nr - torsion. The equiv alent resu lt for CK 1  M r ( D )  will fo llo w from the previous lemma. F or a ∈ CK 1  M r ( D )  = D ∗ /F ∗ r D ′ , w e ha ve a ∈ D ∗ with a n = Nrd D ( a ) d a for d a ∈ D ′ (b y Lemma 2.2.1 ). Since Nrd D : D ∗ → F ∗ , Nrd D ( a ) ∈ F ∗ and therefore a n ∈ F ∗ D ′ . The n a nr ∈ F ∗ r D ′ , which sho ws a nr = 1, pro ving CK 1  M r ( D )  is also nr -torsion. Let F be a ﬁxed ﬁeld. As fo r K 0 , we will observ e that CK 1 and ZK 1 form functors from the category of central simple algebras o v er F to the category of Chapter 2. Algebraic K - Theory 50 ab elian groups. F or an y t w o cen tral simple algebras A and B ov er F with an F - algebra homomorphism φ : A → B , there is a comm utative diagram: 1 / / ZK 1 ( A )   / / K 1 ( F ) id   / / K 1 ( A ) φ   / / CK 1 ( A )   / / 1 1 / / ZK 1 ( B ) / / K 1 ( F ) / / K 1 ( B ) / / CK 1 ( B ) / / 1 where φ : K 1 ( A ) → K 1 ( B ) is deﬁned b y φ  ( a ij ) E ( A )  =  φ ( a ij )  E ( B ) for each ( a ij ) E ( A ) ∈ K 1 ( A ). It follows that CK 1 and ZK 1 are functors from the category of cen tral simple algebras ov er F to the category of ab elian groups. W e also o bserv e tha t CK 1 do es no t resp ect Morita inv ariance. F o r a division algebra D ov er the ﬁeld F , the exact sequence ( 2.3 ) can b e written as 1 − → D ′ ∩ F ∗ − → F ∗ − → D ∗ /D ′ − → D ∗ /F ∗ D ′ − → 1 since the ma p K 1 ( F ) → K 1 ( D ) tak es ( a ij ) E ( F ) to ( a ij ) E ( D ). F ro m sequence ( 2.4 ) w e sa w that CK 1 ( M r ( D )) = D ∗ /F ∗ r D ′ , but CK 1 ( D ) = D ∗ /F ∗ D ′ . In general they are not isomorphic (see for example [ 33 , Eg. 7]). Let us a lso men t io n a no ther group which exhibits some similar prop erties to CK 1 ( D ). F or a cen tral simple a lgebra A o v er a ﬁeld F , t he gr oup G ( A ) = A ∗ / ( A ∗ ) 2 , called the square class group of A , has b een studied b y Lewis and Tignol [ 45 ]. They note that the group G ( A ) is a torsion ab elian group of exp onent t wo. They sho w that when A is a cen tral simple alg ebra of o dd degree ov er a ﬁeld F , then the map G ( F ) → G ( A ) induced by inclusion is an isomorphism (see [ 45 , Cor. 2, p. 367]). Although, in some asp ects, the b ehaviour of G ( A ) is similar to that of CK 1 , [ 45 , Prop. 5] sho ws an a sp ect where they diﬀer. It sa ys that if D is a division ring and n is a p ositive in teger greater than 2, then G ( M n ( D )) ∼ = G ( D ), which w e observ ed ab ov e do es not hold for CK 1 ( D ). Chapter 2. Algebraic K - Theory 51 2.3 Higher K -The o ry F or higher algebraic K -theory , the K -groups were deﬁned by Quillen in the early 1970s. Quillen ga v e t w o diﬀeren t constructions for higher K - theory , called the +- construction and the Q - construction. The +-construction deﬁnes the higher K - groups of a ring R . The Q -construction deﬁnes the K - groups of an exact category and, for a ring R , the K -gro ups K i ( R ) are deﬁned to b e the K -g roups K i ( P r( R )) where P r( R ) is the category of ﬁnitely g enerated pro jectiv e R - mo dules. The tw o constructions do in fa ct giv e the same K -groups for a ring R , although in app earance they are very diﬀeren t. (The pro of is v ery inv olved , see [ 70 , § IV.7].) F or i = 0, 1, 2 the construction a g rees whic h the deﬁnitions giv en in Section 2.1 (see [ 58 , § 5.2.1]). The K -groups, although complicated to deﬁne, are f unctorial in construction. W e recall b elo w some o f their basic prop erties. Example 2.3.1. (See [ 58 , Thm. 5.3.2].) Let F q b e a ﬁnite ﬁeld with q elemen ts. Then K 0 ( F q ) = Z and for i ≥ 1, K i ( F q ) ∼ =      Z q n − 1 if i = 2 n − 1 , 0 if i is ev en. Theorem 2.3.2. I f F : C → D is an exact functor b etwe en exact c ate gories, then F induc es a map F ∗ : K i ( C ) → K i ( D ) . In p a rticular, e ac h K i is a functor fr om the c ate gory of e xact c ate gorie s with exact functors to the c ate go ry of ab el i a n gr oups. Mor e o v er, isom orphic functors i n duc e the same m a p on the K -gr oups. Pro of. See [ 70 , p. IV.51], [ 54 , p. 19].  Corollary 2.3.3. F or i ≥ 0 , e ach K i is a f unc tor fr om the c ate gory of rings to the c ate gory of ab elian gr oups. Pro of. See [ 70 , § IV.1.1.2] or [ 58 , Eg. 5.3.22]. This follo ws from Theorem 2.3.2 , s ince a ring homomorphism φ : R → S induces an exact functor S ⊗ R − : P r( R ) → P r( S ).  Chapter 2. Algebraic K - Theory 52 Theorem 2.3.4. If the rings R and S ar e Morita e quivale nt, then K i ( R ) ∼ = K i ( S ) for e ach i ≥ 0 . Pro of. See [ 70 , § IV.6.3.5]. This follows fro m Theorem 2.3 .2 , since if R and S are Morita equiv alen t then t here is an equiv alence of categories P r( R ) ∼ = P r( S ). It follo ws that K i ( R ) ∼ = K i ( S ) for eac h i ≥ 0.  Let C and D b e exact categories. The category of f unctors from C to D is an exact category whic h is denoted by [ C , D ] and with morphisms deﬁned to b e nat ur a l transformations. Then b y [ 54 , p. 22], a sequence of functors from C to D 0 − → F ′ − → F − → F ′′ − → 0 is an exact sequenc e of exact functors if for all A ∈ C , 0 − → F ′ ( A ) − → F ( A ) − → F ′′ ( A ) − → 0 is an exact sequenc e in D . Theorem 2.3.5. L et C and D b e exact c ate gories a nd let 0 − → F ′ − → F − → F ′′ − → 0 b e an exact se quenc e of exa c t func tors fr om C to D . Then F ∗ = F ′ ∗ + F ′′ ∗ : K i ( C ) − → K i ( D ) . Pro of. See [ 54 , Cor. 1, p. 22].  In the a b ov e t heorem, suppose C = D is the category of ﬁnitely generated pro- jectiv e mo dules (or the category of graded ﬁnitely generated pro jective modules: see Section 4.5 for its deﬁnition). T ake b oth F ′ and F ′′ to b e the iden tit y functor and Chapter 2. Algebraic K - Theory 53 let F : C → C ; A 7→ A ⊕ A . Then clearly 0 − → A ı − → A ⊕ A π − → A − → 0 is an exact sequence, with maps ı : A → A ⊕ A ; x 7→ ( x, 0) and π : A ⊕ A → A ; ( x, y ) 7→ y . So the homomorphism F ∗ : K i ( C ) → K i ( C ) induced b y F is F ∗ = id + id : K i ( C ) − → K i ( C ) a 7− → a + a. By induction, if F : C → C ; A 7→ A k for k ∈ N , then the induced homomorphism is F ∗ : K i ( C ) → K i ( C ); a 7→ k a , whic h is also m ultiplication b y k . W e will use this result in the pro ofs of Prop ositions 3.1.3 and 5.4.3 . The follow ing r esult was pr ov en b y Green et al. [ 24 ]. Theorem 2.3.6. F o r a ring R , let f : R → R b e an inn e r automorphism of R with f ( r ) = a − 1 r a , wher e a is a unit of R . The n K i ( f ) : K i ( R ) → K i ( R ) is the identity. Pro of. (See [ 24 , Lemma 2].) Since f : R → R is a ring homomorphism, there is a functor F : P r( R ) − → P r( R ); P 7− → R ⊗ f P , where R is a rig ht R -mo dule via f and w e note that r ⊗ r ′ p = r f ( r ′ ) ⊗ p . W e will sho w that there is a natural isomorphism φ f rom the iden tity functor to the functor F . F or P ∈ P r( R ), deﬁne φ P : P − → R ⊗ f P ; p 7− → 1 ⊗ ap. Note that φ P is an R -mo dule homomorphism, since φ P ( r p ) = 1 ⊗ ar p = 1 ⊗ ar a − 1 ap = 1 ⊗ f − 1 ( r ) ap = r ⊗ ap = r ( φ P ( p )) . Chapter 2. Algebraic K - Theory 54 Then for an R -mo dule homomorphism g : P → P ′ in P r( R ), the follow ing diagr a m comm utes: P id( g )   φ P / / R ⊗ f P F ( g )=1 ⊗ g   P ′ φ P ′ / / R ⊗ f P ′ So φ forms a natural transformation from t he iden tit y f unctor to F . Each φ P is an isomorphism in P r( R ) since the map φ − 1 P : R ⊗ f P − → P ; r ⊗ p 7− → r a − 1 p is an R -mo dule homomorphism which is an inv erse of φ P . So the inv erses φ − 1 P form a natura l transformation whic h is an in v erse of φ . By applying Theorem 2.3.2 , this sho ws that the induced map K i ( R ) → K i ( R ) is the iden tit y .  The ab o v e theorem allo w ed Green et al. to pro v e the main theorem of their pap er [ 24 , Thm. 4], whic h sho ws tha t K i ( D ) ⊗ Z [1 /n ] ∼ = K i ( F ) ⊗ Z [1 /n ] for a division algebra D ov er F of dimension n 2 . In Chapter 3 , w e will generalise this result to co v er Azuma y a a lgebras which are free o v er their cen tres. Their pro of uses t ha t fact that K i ( R ) → K i ( M t ( R )) → K i ( R ) is multiplication by t , where R → M t ( R ) is the dia g onal homomorphism r 7→ r I t (see [ 24 , Lemma 1]), and also the Sk olem-No ether theorem whic h gua r an tees that algebra homomorphisms in t he setting of cen tral simple algebras are inner auto morphisms. Their pro of combine s these results with Theorem 2.3.6 and with the main result of [ 16 ] whic h states t ha t lim i →∞ M n 2 i ( F ) ∼ = lim i →∞ M n 2( i +1) ( D ) . Chapter 3 K -Theory of Azuma y a Algebras As w e noted in the previous c hapter, Green et al. [ 24 ] prov ed that for a division algebra ﬁnite dimensional ov er its cen tre, its K -theory is “essen tia lly the same” as the K -theory of its c entre; that is, for a divis ion alg ebra D o ve r its cen tre F of index n , K i ( D ) ⊗ Z [1 /n ] ∼ = K i ( F ) ⊗ Z [1 /n ] . (3.1) In this c hapter, w e pro v e that the isomorphism ( 3.1 ) holds for an y Azuma y a algebra free o v er its cen tre (see Theorem 3.1.5 ). A corollary of this is that the isomorphism holds for Azumay a a lgebras o v er semi-lo cal rings. This extends the res ults of Hazrat (see [ 27 , 28 ]) where ( 3.1 ) ty p e pro p erties ha v e b een prov en for cen tral simple algebras and Azuma y a algebras o ve r lo cal rings resp ectiv ely . In Section 3.1 , w e introduce an abstract functor called a D -functor, whic h is deﬁned on the category of Azuma ya a lgebras free ov er a ﬁxed base ring. This is a con tin uation o f [ 27 ] and [ 28 ] where Hazrat deﬁnes similar functors, o v er categories of cen tral simple algebras and Azuma ya algebras respectiv ely . Here w e show in Theorem 3.1.2 that the range of a D -functor is the category of b ounded torsion ab elian groups. W e then prov e that the k ernel and cok ernel of the K -groups are D - functors, whic h allo ws us to prov e ( 3.1 ) t yp e pro p erties for Azuma y a algebras whic h are free o v er their centres (see Theorem 3.1.5 and [ 31 , Thm. 6]). The Ho c hsc hild homology of Azumay a alg ebras b eha ve s in a similar wa y to the 55 Chapter 3. K -Theory o f Azuma y a Algebras 56 K -theory of Azuma ya algebras. Corti ˜ nas and W eib el [ 13 ] ha v e show n that there is a similar result to Theorem 3.1.5 for the Ho chs child homology of an Azumay a algebra. In Section 3.2 , we b egin b y recalling the deﬁnition of Ho c hsc hild homology , whic h can b e found in [ 47 , 71 ]. W e then recall the result of Corti ˜ nas and W eib el whic h sho ws that H H k ∗ ( A ) ∼ = H H k ∗ ( R ) for an R -Azumay a alg ebra A of constan t rank. 3.1 D -functors Throughout this section, let R b e a ﬁxed comm utativ e ring and A b b e the category of ab elian groups. Let Az( R ) b e the category o f Azuma y a alg ebras free o v er R with R -algebra homomorphisms. Deﬁnition 3.1.1. Consider a functor F : Az( R ) → A b ; A 7→ F ( A ). Suc h a functor is called a D -functor if it satisﬁes the follo wing three prop erties: (1) F ( R ) is the trivial gro up, (2) F or an y Azumay a algebra A free ov er R and any k ∈ N , there is a ho momorphism ρ : F ( M k ( A )) − → F ( A ) suc h that the comp o sition F ( A ) → F ( M k ( A )) → F ( A ) is η k , where η k ( x ) = x k . (3) With ρ as in pr o p ert y (2), then k er( ρ ) is k - torsion. The name D -functor comes from D ieudonn ´ e, since Dieudonn ´ e determinan t b e- ha v es in a similar w a y to Prop erty (2 ). Note that M k ( A ) is an Azumay a alg ebra free ov er R b y Prop osition 1.4.6 , since A and M k ( R ) are Azuma ya algebras free o v er R (see Example 1.4.5 ) and M k ( A ) ∼ = A ⊗ R M k ( R ). Also note that t he natural R -algebra homomorphism A → M k ( A ); a 7→ aI k induces a group ho momorphism F ( A ) → F ( M k ( A )). The follo wing theorem sho ws that the range of a D -functor is a b ounded to rsion ab elian group. Chapter 3. K -Theory o f Azuma y a Algebras 57 Theorem 3.1.2. L et A b e an Azumaya algebr a fr e e over R of dimension n and let F b e a D -functor. The n F ( A ) is n 2 -torsion. Pro of. W e b egin b y applying the deﬁnition of a D -functor to the R -Azuma ya a l- gebra R . Prop ert y (1) sa ys that F ( R ) is the trivial gro up and prop ert y ( 2 ) says that the comp osition F ( R ) → F ( M n ( R )) → F ( R ) is η n . So b y the third pro p- ert y , F ( M n ( R )) is n - torsion. Since A is an Azuma ya algebra free ov er R , w e ha ve A ⊗ R A op ∼ = End R ( A ) ∼ = M n ( R ), whic h means tha t F ( A ⊗ R A op ) ∼ = F ( M n ( R )) is n -torsion. The natural R -algebra homomorphisms A → A ⊗ R A op , A op → End R ( A op ), End R ( A op ) ∼ = M n ( R ) and A ⊗ R M n ( R ) ∼ = M n ( A ) com bine to g iv e the following R -algebra homomorphisms A − → A ⊗ R A op − → A ⊗ End R ( A op ) − → A ⊗ M n ( R ) − → M n ( A ) . These induce the homomorphisms F ( A ) i → F ( A ⊗ R A op ) r → F ( M n ( A )). By prop- ert y (2) in the deﬁnition of a D - functor, we ha v e a homomorphism ρ : F ( M n ( A )) → F ( A ) suc h that the comp osition F ( A ) → F ( M n ( A )) → F ( A ) is η n . Conside r the follo wing diagr a m F ( A ) i   η n   F ( A ⊗ R A op ) r   F ( M n ( A )) ρ / / F ( A ) (3.2) whic h is comm utative b y prop erty (2). Let a ∈ F ( A ). Then a n 2 =  η n ( a )  n = ρ ◦ r  ( i ( a )) n  = ρ ◦ r (1) = 1 since F ( A ⊗ A op ) is n -torsion. Th us F ( A ) is n 2 -torsion.  Chapter 3. K -Theory o f Azuma y a Algebras 58 F or a ring A with cen tre R , consider the inclusion R → A . By Corollary 2.3.3 , this induces the map K i ( R ) → K i ( A ) for i ≥ 0. Consider the exact sequence 1 − → ZK i ( A ) − → K i ( R ) − → K i ( A ) − → CK i ( A ) − → 1 (3.3) where ZK i and CK i are the k ernel and cokerne l of the map K i ( R ) → K i ( A ) resp ec- tiv ely . W e will observ e that CK i can b e considered as the following functor CK i : Az ( R ) − → A b A 7− → CK i ( A ) . F or an Azuma y a a lg ebra A free o v er R , clearly CK i ( A ) = cok er  K i ( R ) → K i ( A )  is an ab elian group. F or a homomorphism f : A → A ′ of R - Azuma y a algebras, there is a n induced group homomorphism f ∗ : K i ( A ) → K i ( A ′ ). The map f restricted to R is the identit y map, so it induces the iden tity on the lev el of the K - groups. Since K i preserv es comp ositions o f maps, the follo wing diagram is comm utative K i ( R ) / / id   K i ( A ) / / f ∗   CK i ( A ) CK i ( f )   K i ( R ) / / K i ( A ′ ) / / CK i ( A ′ ) Then it can be easily che ck ed that CK i forms the required functor. Similarly , w e can consider ZK i as the functor ZK i : Az ( R ) − → A b A 7− → ZK i ( A ) . Prop osition 3.1.3. With CK i deﬁne d as ab ove, CK i is a D -func tor. Pro of. Prop erty (1) in the deﬁnition of a D - f unctor is clear, since K i ( Z ( R )) → K i ( R ) is the iden tity map, and so clearly CK i ( R ) is trivial. Chapter 3. K -Theory o f Azuma y a Algebras 59 T o pro v e the second prop ert y , let A b e an Azuma y a algebra free ov er R and let k ∈ N . Consider the functors P r( A ) φ − → P r( M k ( A )) and P r( M k ( A )) ψ − → P r( A ) X 7− → M k ( A ) ⊗ A X Y 7− → A k ⊗ M k ( A ) Y By Theorem 2.3.2 , these functors induce homomorphisms φ and ψ on the lev el o f the K -groups. Morita theory shows that ψ is an equiv alence of categories (see page 39 ), so using Theorem 2.3.4 , it induces a n isomorphism from K i ( M k ( A )) to K i ( A ). F or eac h X ∈ P r( A ), w e hav e ψ ◦ φ ( X ) ∼ = X k . Since K i are func tor s whic h resp ect direct sums (see the remarks after Theorem 2.3.5 ), this composition induces a m ultiplication b y k on the lev el of the K -groups. W e hav e the following comm uta t ive diagram K i ( R ) f / / id   K i ( A ) f / / φ   CK i ( A ) / /   1 K i ( R ) g / / η k   K i ( M k ( A )) g / / ψ ∼ =   CK i ( M k ( A )) / / ρ   1 K i ( R ) f / / K i ( A ) f / / CK i ( A ) / / 1 (3.4) where comp ositions of columns are η k , and th us prop ert y (2) holds. No w let x ∈ CK i ( M k ( A )) suc h that ρ ( x ) = 1. Then there is a ∈ K i ( M k ( A )) with g ( a ) = x so that 1 = f ◦ ψ ( a ). As the ro ws are exact, there is b ∈ K i ( R ) with f ( b ) = ψ ( a ). T aking p ow ers of k , w e hav e f ( b k ) = ψ ( a k ) and g ( a k ) = x k . As ψ is an isomorphism and f ◦ η k ( b ) = ψ ◦ g ( b ), it follow s tha t a k = g ( b ). Then b y the exactness of t he row s, g ◦ g ( b ) = 1; tha t is, x k = 1. So CK i satisﬁes prop erty (3) of a D -functor.  Prop osition 3.1.4. With ZK i deﬁne d as ab ove, ZK i is a D -func tor. Pro of. This follow s in exactly the same w a y as Prop osition 3.1.3 .  W e are no w in a p osition to prov e that the K -theory o f Azuma ya algebras free Chapter 3. K -Theory o f Azuma y a Algebras 60 o v er their cen tres and the K - theory of Azuma ya algebras o v er semi-lo cal rings a re isomorphic to the K -theory of their cen tres up to (their ranks) torsions. Theorem 3.1.5. L e t A b e an Azumaya alg ebr a fr e e over its c entr e R of dim e nsion n . Then for any i ≥ 0 , K i ( A ) ⊗ Z [1 /n ] ∼ = K i ( R ) ⊗ Z [1 /n ] . Pro of. Prop o sitions 3.1.3 and 3 .1.4 sho w that CK i and ZK i are b o t h D -functors. By The orem 3.1.2 , it follo ws that CK i ( A ) and ZK i ( A ) are n 2 -torsion ab elian groups. T ensoring the exact sequence ( 3.3 ) by Z [1 /n ], since CK i ( A ) ⊗ Z [1 /n ] and ZK i ( A ) ⊗ Z [1 /n ] v anish, the result follows.  W e recall the deﬁnition of a pro jectiv e mo dule of constan t rank, whic h is used in the corollary b elow . Deﬁnition 3.1.6. Let R b e a commutativ e ring and let P b e a prime ideal of R . Set S = R \ P and write R P for the lo calisation S − 1 R , which is a lo cal ring by [ 48 , Ex. 6.45(i)]. (Recall a ring R is lo c al if the non-in v ertible elemen ts of R constitute a prop er 2- sided ideal of R .) F or an R -mo dule M , w e write M P = S − 1 M . If M is a ﬁnitely generated pro jectiv e R -mo dule, then, b y [ 48 , Prop. 6.4 4], M P is a ﬁnitely generated pro jectiv e mo dule ov er R P . E ve ry ﬁnitely generated pro jectiv e mo dule o v er a lo cal ring is free with a uniquely deﬁned rank (see [ 58 , Thm. 1.3.11]). Then rank P ( M ) is deﬁned to b e the rank of M P as an R P -mo dule. W e sa y tha t M is of c onstant r ank if r ank P ( M ) = n is the same for all prime ideals P and w e write rank( M ) = n . If M is a free mo dule o v er the ring R , then M is of constan t ra nk since M P ∼ = R P ⊗ R M (see [ 48 , Prop. 6.55]) and dim R ( M ) = dim R P ( R P ⊗ R M ) = rank( M ). Corollary 3.1.7. L et R b e a semi-lo c al ring and let A b e an Azumaya algebr a o v er its c entr e R of r ank n . Then for any i ≥ 0 , K i ( A ) ⊗ Z [1 /n ] ∼ = K i ( R ) ⊗ Z [1 /n ] . Chapter 3. K -Theory o f Azuma y a Algebras 61 Pro of. Since A is ﬁnitely generated pro jectiv e of constan t rank and R is a semi-local ring, it follo ws that A is a free mo dule o v er R (see [ 8 ], § I I.5.3, Prop. 5), and th us the corollary follows from Theorem 3.1.5 .  Theorem 3.1.5 co v ers Azuma y a algebras whic h are free ov er their cen tres. W e also men tion here a theorem of Ha zrat, Ho obler [ 29 , Thm. 12], whic h sho ws t ha t a similar result holds for the K -theory of Az umay a algebras ov er shea v es. Their result co v ers the case of Azuma ya algebras o v er No etherian cen tres, but it remains as a question whether this result holds for any Azuma y a algebra of constant rank. Question 3.1.8. Let A b e an Azuma y a a lgebra o v er its cen t re R of constan t ra nk n . Then is it true that for an y i ≥ 0, K i ( A ) ⊗ Z [1 /n ] ∼ = K i ( R ) ⊗ Z [1 /n ]? 3.2 Homology of Azu ma y a alge bras W e b egin this section by recalling the deﬁnition of Ho c hsc hild homolo gy , whic h can b e found in [ 47 , 71 ]. Let R b e a ring. Recall that a c hain complex o f R -mo dules ( C ∗ , d ∗ ) is a family of righ t R -mo dules { C n } n ∈ Z together with R -mo dule homomorphisms d n : C n → C n − 1 , suc h tha t the comp osition of any t w o consecutiv e maps is zero: d n ◦ d n +1 = 0 for all n . The maps d n are called the diﬀeren tials of C ∗ . The chain complex is usually written as: · · · − → C n +1 d n +1 − → C n d n − → C n − 1 d n − 1 − → C n − 2 − → · · · The k ernel of d n is denoted b y Z n , and the image of d n +1 is denoted b y B n . Then for all n , 0 ⊆ B n ⊆ Z n ⊆ C n . The n th homology mo dule of C ∗ is the quotien t H n ( C ∗ ) = Z n /B n . Let K b e a comm utativ e ring, R b e a K -a lgebra, and M an R -bimo dule. W e will write R ⊗ n for the n - f old tensor pro duct of R ov er K . The c hain complex whic h Chapter 3. K -Theory o f Azuma y a Algebras 62 giv es rise to Ho chs c hild homolog y is given b y C ∗ = M ⊗ R ⊗∗ and is written as · · · − → M ⊗ R ⊗ n d n − → M ⊗ R ⊗ n − 1 d n − 1 − → · · · d 2 − → M ⊗ R d 1 − → M − → 0 . Its diﬀeren tials are d n = P n i =0 ( − 1) i ∂ i where ∂ i are deﬁned b y ∂ i ( m ⊗ a 1 ⊗ · · · ⊗ a n ) =          ma 1 ⊗ a 2 ⊗ · · · ⊗ a n if i = 0 m ⊗ a 1 ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ a n if 1 < i < n a n m ⊗ a 1 ⊗ · · · ⊗ a n − 1 if i = n for a i ∈ R and m ∈ M . Then by [ 47 , Lemma 1.1 .2] d n ◦ d n +1 = 0, so ( C ∗ , d ∗ ) is a chain complex called the Ho chsc hild complex. The n th Ho chschild homolo gy mo dule of R with co eﬃcien t s in M is deﬁned to b e H n ( M ⊗ R ⊗∗ ) and is denoted by H n ( R, M ). The direc t sum L n ≥ 0 H n ( R, M ) is denoted b y H ∗ ( R, M ), and w e will write H H ∗ ( R ) for H ∗ ( R, R ). By [ 47 , § 1.1.5], if R is comm utative then H H ∗ ( R ) is an R -mo dule. F or a comm utative ring R and an R -algebra A , supp ose that K is a comm uta tiv e subring of R . The n we consider the Ho chs c hild homology , denoted by H H K ∗ , of R and A as K -algebras. The f o llo wing result of Corti ˜ nas and W eib el [ 13 ] show s t ha t for an R -Azuma y a algebra A of constan t rank, its Ho c hsc hild ho mo lo gy b eha v es in a similar wa y to its K -theory (see Theorem 3.1.5 ). Theorem 3.2.1. L et A b e an Azumaya algebr a over a K -alge br a R . If A has c on- stant r ank, then ther e is an isomorphism H H K ∗ ( A ) ∼ = H H K ∗ ( R ) . Pro of. See [ 13 , p. 53].  Chapter 4 Graded Azuma y a Algebras This c hapter con tains some basic deﬁnitions in the graded setting. These deﬁnitions can b e found in [ 14 , 39 , 51 ], though not alw ay s in the generalit y that w e require. W e also include in this c hapter a nu mber o f results in the graded setting whic h will b e used in Chapter 5 . In Section 4.1 w e deﬁne g raded rings and give the graded v ersions of the deﬁnitions of ideals, factor ring s and ring homo mor phisms. In Section 4.2 w e giv e the corresp onding deﬁnitions for graded mo dules. W e then deﬁne graded division rings and sho w that a graded mo dule ov er a graded division ring is graded free with a uniquely deﬁned dimension. In Section 4.3 , we study graded cen tral simple algebras graded b y an arbitrary ab elian group. W e observ e t ha t t he tensor pro duct o f tw o graded cen tral simple R -algebras is graded cen tral simple (Prop ositions 4.3.1 and 4.3 .2 ). This result has b een pro v en b y W a ll for Z / 2 Z -graded cen tral sim ple algebras (see [ 72 , Thm. 2]), and b y Hwang and W a dsw orth fo r R -algebras with a tota lly ordered ab elian, a nd hence torsion-free, grade group (see [ 39 , Prop. 1.1]). W e then observ e that a gr aded central simple algebra, graded b y an a b elian gr oup, is an Az umay a algebra (T heorem 4.3.3 ). This result extends the result of Boulagouaz [ 6 , Prop. 5.1 ] a nd Hwang, W a dsw o rth [ 39 , Cor. 1.2] ( f or a to tally ordered a b elian gra de group) to gra ded rings in whic h the grade group is not totally ordered. W e deﬁne grading on matrices in Section 4.4 , and observ e some prop erties of 63 Chapter 4. G r a ded Azuma y a Alg ebras 64 these graded matrix ring s. W e generalise a result of Caenep eel et a l. [ 11 , Thm. 2.1] in Theorem 4.4.9 . W e ha v e also rewritten a n um b er of kno wn results on simple rings in the gra ded setting, whic h w ere required for the pro of of t his theorem. In Section 4.5 , w e deﬁne graded pro jectiv e mo dules and graded Azuma y a alg ebras. W e pro v e tha t for a graded ring R , the graded matrix ring ov er R is Morita equiv a len t to R (see Prop osition 4.5.4 ). 4.1 Graded rings A ring R = L γ ∈ Γ R γ is called a Γ -gr a de d ring , or simply a gr ade d rin g , if Γ is a group, eac h R γ is an additiv e subgroup of R and R γ · R δ ⊆ R γ δ for all γ , δ ∈ Γ. W e remark that initially Γ is an arbitrar y group whic h is not necessarily ab elian, so we will write Γ a s a m ultiplicativ e gro up with iden tit y elemen t e . Eac h x ∈ R can b e uniquely expressed as a ﬁnite sum x = P γ ∈ Γ x γ with each x γ ∈ R γ . F or eac h γ ∈ Γ, the elemen ts of R γ are said to b e homo gene o us of de gr e e γ and we write deg ( r ) = γ if r ∈ R γ . W e let R h = S γ ∈ Γ R γ b e the set of homogeneous elemen ts of R . The set Γ R =  γ ∈ Γ : R γ 6 = { 0 }  , whic h is also denoted by Su pp ( R ), is called the supp ort (or grade se t) of R . W e note that the suppo rt of R is not necessarily a g r o up. Examples 4.1.1. 1. Let ( Γ , · ) be a group and R b e a ring. Set R = L γ ∈ Γ R γ where R e = R and for a ll γ ∈ Γ with γ 6 = e , let R γ = 0. Then R can b e considered as a trivially Γ-graded ring, with Supp( R ) = { e } . 2. Let A b e a ring. Then the p olynomial ring R = A [ x ] is a Z -graded ring, with R = L n ∈ Z R n where R n = Ax n for n ≥ 0 and R n = 0 fo r n < 0. Here Supp( R ) = N ∪ { 0 } . 3. Let ( G , · ) b e a group and let A b e a ring . Then the group ring R = A [ G ] is graded ring, with R = L g ∈ G R g where R g = { ag : a ∈ A } , and Supp ( R ) = G . Chapter 4. G r a ded Azuma y a Alg ebras 65 Prop osition 4.1.2. L et R = L γ ∈ Γ R γ b e a Γ -gr ade d ring. The n : 1. 1 R is homo ge ne ous of de gr e e e , 2. R e is a subring of R , 3. Each R δ is an R e -bimo dule, 4. F o r an invertible element r ∈ R δ , its inverse r − 1 is homo gene ous of de gr e e δ − 1 . Pro of. (1) Supp ose 1 R = P γ ∈ Γ r γ for r γ ∈ R γ . F or some δ ∈ Γ, let s ∈ R δ b e an arbitrary non-zero elemen t . Then s = s 1 R = P γ ∈ Γ sr γ where sr γ ∈ R δγ for all γ ∈ Γ. The decomp osition is unique, so sr γ = 0 for all γ ∈ Γ with γ 6 = e . But as s w as arbitrary , this holds for all s ∈ R (not necessarily homogeneous), and in particular 1 R r γ = r γ = 0 if γ 6 = e . F or γ = e , w e ha v e 1 R = r e , so 1 R ∈ R e . (2) This follows since R e is a n additiv e subgroup of R with R e R e ⊆ R e and 1 ∈ R e . (3) This is immediate. (4) Let x = P γ x γ (with deg( x γ ) = γ ) b e the in v erse o f r , so that 1 = r x = P γ r x γ where r x γ ∈ R δγ . Since 1 is homogeneous of degree e and the decomposition is unique, it follows that r x γ = 0 for a ll γ 6 = δ − 1 . Since r is in v ertible, w e ha v e x δ − 1 6 = 0, so x = x δ − 1 as required.  W e sa y that a Γ-graded ring R = L γ ∈ Γ R γ is a str ongly gr a d e d ring if R γ R δ = R γ δ for all γ , δ ∈ Γ. A gra ded ring R is called a cr osse d pr o duct if there is an in ve rtible elemen t in ev ery homogeneous comp onent R γ of R ; that is, R ∗ ∩ R γ 6 = ∅ for all γ ∈ Γ. F or a Γ-graded ring R , let R h ∗ b e the set of in v ertible homogeneous elemen ts of R . Then R h ∗ is a subgroup of R ∗ , and the degree map deg : R h ∗ → Γ is a group homomorphism. Let Γ ∗ R =  γ ∈ Γ : R ∗ ∩ R γ 6 = ∅  , b e the supp ort o f the inv ertible ho mogeneous elemen t s of R . W e note that R is a crossed pro duct if and only if Γ ∗ R = Γ, whic h is equiv alen t to the degree map b eing surjectiv e. Chapter 4. G r a ded Azuma y a Alg ebras 66 Prop osition 4.1.3. L et R = L γ ∈ Γ R γ b e a Γ -gr ade d ring. The n : 1. R is str ongly gr ade d if and on ly i f 1 ∈ R γ R γ − 1 for al l γ ∈ Γ , 2. R is a cr osse d pr o duct if and only if the de gr e e map is surje ctive, 3. If R is a cr osse d pr o duct, then R is a str ongly gr ade d ring. Pro of. (1) The forward direction is immediate. Supp ose 1 ∈ R γ R γ − 1 for all γ ∈ Γ. Then for σ , δ ∈ Γ , R σδ = R e R σδ = ( R σ R σ − 1 ) R σδ = R σ ( R σ − 1 R σδ ) ⊆ R σ R δ pro ving R σδ = R σ R δ , so R is strongly gra ded. (2) This is immediate. (3) F or δ ∈ Γ, there exists r ∈ R ∗ ∩ R δ . So r − 1 ∈ R δ − 1 and 1 = r r − 1 ∈ R δ R δ − 1 .  A t w o- sided ideal I of R is called a homo g ene ous id e al ( or gr ade d ide al ) if I = M γ ∈ Γ ( I ∩ R γ ) . There are s imilar no t io ns of graded subring, graded left ideal a nd graded right ideal. When I is a homog eneous ideal of R , the factor ring R/I forms a graded ring, with R/I = M γ ∈ Γ ( R/I ) γ where ( R/I ) γ = ( R γ + I ) /I . A g raded ring R is s aid to be gr ade d s i m ple if the o nly homogeneous tw o-sided ideals of R are { 0 } and R . Prop osition 4.1.4. An ide al I of a gr ade d ring R is a homo gene ous ide al if and only if I is gener ate d as a two-s i d e d ide al of R by homo gene ous elements. Chapter 4. G r a ded Azuma y a Alg ebras 67 Pro of. Supp ose I = L γ ∈ Γ ( I ∩ R γ ), and consider I ∩ R h . This is a subset of I consisting o f homogeneous elemen ts. Any x ∈ I can b e written a s x = P γ ∈ Γ x γ where x γ ∈ I ∩ R γ . So x γ ∈ I ∩ R h , and th us x can b e written a s a sum of elemen ts of I ∩ R h . Con v ersely , supp ose I is generated b y the set { x j } j ∈ J ⊆ I consisting of homo- geneous elemen ts, for some indexing set J . Then for an y x ∈ I , we can write x as x = P r k x j s l where r k , s l ∈ R h . Then for some r k , x j , s l with α = deg ( r k x j s l ), w e ha v e r k x j s l ∈ I ∩ R α , since x j ∈ I . This sho ws that I = L γ ∈ Γ ( I ∩ R γ ), as required.  Let R and S b e Γ-gr aded r ing s. Then a gr ade d ring homomorphis m f : R → S is a ring homomorphism suc h that f ( R γ ) ⊆ S γ for all γ ∈ Γ. F urther, f is called a gr ade d isomorphism if f is bijectiv e a nd, when suc h a graded isomorphism exists, w e write R ∼ = gr S . W e remark that if f is a graded ring homomorphism whic h is bijectiv e, then its inv erse f − 1 is also a graded ring homomorphism. F or a graded r ing homomorphism f : R → S , w e kno w from the no n- graded setting [ 37 , p. 122] that k er( f ) is an ideal of R a nd im( f ) is a subring of S . It can easily b e sho wn that k er ( f ) is a graded ideal of R and im( f ) is a g raded subring of S . Note that if Γ is an ab elian group, then the cen tre Z ( R ) of a graded ring R is a graded subring of R . If Γ is not ab elian, then the cen t r e of R ma y not b e a gr a ded subring of R , as is sho wn by the fo llowing example. Example 4.1.5. Let G = S 3 = { e, a, b, c, d, f } b e the symmetric group of order 3, where a = (23) , b = (13) , c = (12 ) , d = (123) , f = (132) . Let A b e a ring, a nd consider the group ring R = A [ G ], which is a G -graded ring b y Example 4.1.1 (3). Let x = 1 d + 1 f ∈ R , where 1 = 1 A , and w e note that x is not homogeneous in R . Then x ∈ Z ( R ), but t he homogeneous comp onen ts of x are not in the cen tr e of R . As x is expressed uniquely as the sum o f homogeneous comp onen ts, we ha v e x / ∈ L g ∈ G ( Z ( R ) ∩ R g ). Chapter 4. G r a ded Azuma y a Alg ebras 68 This example can b e generalised b y taking a non-ab elian ﬁnite gro up G with a subgroup N whic h is no r ma l and non-cen tral. Let A b e a ring and consider the group ring R = A [ G ] as ab ov e. Then x = P n ∈ N 1 n is in the cen tre of R , but the homogeneous comp onents of x are not all in the cen tre of R . 4.2 Graded mo dules Let Γ b e a m ultiplicativ e group a nd let R = L γ ∈ Γ R γ b e a Γ-graded r ing. W e say that the group (Γ , · ) acts fr e e ly (as a left action) on a set Γ ′ if for all γ , γ ′ ∈ Γ, δ ∈ Γ ′ , w e ha v e γ δ = γ ′ δ implies γ = γ ′ , where γ δ denotes the imag e o f δ under the action of γ . A Γ ′ -gr ade d left R -mo dule M is deﬁned to b e a left R -mo dule with a direct sum decomp osition M = L γ ∈ Γ ′ M γ , where eac h M γ is an additiv e subgroup of M and Γ acts freely o n t he set Γ ′ , suc h that R γ · M λ ⊆ M γ λ for all γ ∈ Γ , λ ∈ Γ ′ . F r om now o n , unless otherwi s e state d, a gr ade d mo dule wil l m e an a gr ade d left m o dule. W e remark t ha t here w e deﬁne a graded R - mo dule M whic h is graded b y a set Γ ′ , where the grade group of R a cts freely on Γ ′ . It is also p ossible to ﬁx a group Γ and, fo r all graded rings and mo dules, to w ork instead with this ﬁxed gro up Γ as the gra de group. This is the approac h take n throug ho ut the b o ok [ 51 ]. While our approac h is more general than taking a ﬁxed gro up Γ, in most cases their w eak er approac h suﬃces. As fo r gra ded rings, M h = S γ ∈ Γ ′ M γ and Γ ′ M =  γ ∈ Γ ′ : M γ 6 = { 0 }  . A gr ade d submo dule N of M is deﬁned to be an R -submo dule suc h that N = L γ ∈ Γ ′ ( N ∩ M γ ). F or a graded submo dule N o f M , w e deﬁne the graded quotient structure on M / N b y M / N = M γ ∈ Γ ′ ( M / N ) γ where ( M / N ) γ = ( M γ + N ) / N . A g raded R -mo dule M is said to b e gr ade d simple if t he only graded submodules of M are { 0 } and M . A gr ade d fr e e R -mo dule M is deﬁned to b e a gra ded R -mo dule whic h is free as an R -mo dule with a homogeneous base. Let N = L γ ∈ Γ ′′ N γ b e ano t her g raded R -mo dule, suc h that there is a set ∆ Chapter 4. G r a ded Azuma y a Alg ebras 69 con taining Γ ′ and Γ ′′ as subsets, where Γ acts freely on ∆. The graded R - mo dule M can b e w ritten as M = L γ ∈ ∆ M γ with M γ = 0 if γ ∈ ∆ \ Γ ′ M , and similarly for N . A gr ade d R -mo d ule homom orphism f : M → N is an R -mo dule ho momorphism suc h that f ( M δ ) ⊆ N δ for all δ ∈ ∆. Let Hom R - gr - M od ( M , N ) denote the group o f graded R -mo dule homomorphisms, whic h is an additiv e subgroup of Hom R ( M , N ). If the graded R -mo dule homomorphism f is bijectiv e, then f is called a gr ade d isomorph i s m and we write M ∼ = gr N . If f is a bijectiv e graded R -mo dule homomorphism, then its in v erse f − 1 is also a graded R -mo dule homomorphism. Supp ose ∆ is a group con taining Γ ′ and Γ ′′ as subgroups, where the group Γ acts freely on ∆, and supp ose R , M and N are deﬁned as ab o ve . A graded R -mo dule homomorphism from M to N may also shift the grading on N . F or eac h δ ∈ ∆, w e ha v e a subgroup of Ho m R ( M , N ) of δ - shifted homomorphisms Hom R ( M , N ) δ = { f ∈ Hom R ( M , N ) : f ( M γ ) ⊆ N γ δ for all γ ∈ ∆ } . F or some δ ∈ ∆, w e deﬁne the δ -shifted R - mo dule M ( δ ) a s M ( δ ) = L γ ∈ ∆ M ( δ ) γ where M ( δ ) γ = M γ δ . Then Hom R ( M , N ) δ = Hom R - gr - M od ( M , N ( δ )) = Hom R - gr - M od ( M ( δ − 1 ) , N ) . Let HOM R ( M , N ) = L δ ∈ ∆ Hom R ( M , N ) δ . Let M , N b e ∆-graded R -mo dules as ab ov e. The n M ⊕ N forms a ∆-graded R -mo dule with M ⊕ N = M γ ∈ ∆ ( M ⊕ N ) γ where ( M ⊕ N ) γ = M γ ⊕ N γ . With R deﬁned as ab ov e and f or ( d ) = ( δ 1 , . . . , δ n ) ∈ Γ n , consider R n ( d ) = R ( δ 1 ) ⊕ · · · ⊕ R ( δ n ) = M γ ∈ Γ ( R ( δ 1 ) γ ⊕ · · · ⊕ R ( δ n ) γ ) Chapter 4. G r a ded Azuma y a Alg ebras 70 where R ( δ i ) γ = R γ δ i is the γ -comp onen t of the δ i -shifted g raded R -mo dule R ( δ i ). Note that for eac h i , with 1 ≤ i ≤ n , the elemen t e i of the standard basis fo r R n ( d ) is homogeneous of degree δ − 1 i . Similarly for { δ i } i ∈ I where I is a n indexing set and δ i ∈ Γ, consider the Γ-graded R -mo dule L i ∈ I R ( δ i ). Ag a in the i -th basis elemen t e i of L i ∈ I R ( δ i ) is homogeneous of degree δ − 1 i . Supp ose F is a Γ-graded R -mo dule whic h is graded free with a homogeneous base { b i } i ∈ I , where deg( b i ) = δ i . Then the map ϕ : M i ∈ I R ( δ − 1 i ) − → F e i 7− → b i is a graded R -mo dule isomorphism, and w e write F ∼ = gr L i ∈ I R ( δ − 1 i ). Con v ersely , supp ose F is a Γ-gr a ded R -mo dule with L i ∈ I R ( δ − 1 i ) ∼ = gr F , as graded R -mo dules, for some { δ i } i ∈ I with δ i ∈ Γ. Since { e i } i ∈ I forms a ho mo g eneous basis of L i ∈ I R ( δ − 1 i ), the images of the e i under the graded isomorphism form a homogeneous basis for F . Th us F is graded free if and only if F ∼ = gr L i ∈ I R ( δ − 1 i ) for some { δ i } i ∈ I with δ i ∈ Γ. In t he same w a y , F is g raded free with a ﬁnite basis if and only if F ∼ = gr R n ( d ) for some ( d ) = ( δ 1 , . . . , δ n ) ∈ Γ n . Prop osition 4.2.1. L et R b e a Γ -gr ade d ring and let δ ∈ Γ . Then R ( δ ) ∼ = gr R as gr ade d R -mo d ules if and only if δ ∈ Γ ∗ R . Pro of. If δ ∈ Γ ∗ R , then let x ∈ R ∗ ∩ R δ . Then there is a graded R -mo dule isomor- phism R x : R → R ( δ ); r 7→ r x . Con v ersely , if φ : R ∼ = gr R ( δ ), then φ (1) ∈ R δ with in v erse φ − 1 (1), so φ (1) ∈ R ∗ ∩ R δ .  W e no t e that it f ollo ws from the ab ov e prop osition that R ( δ ) ∼ = gr R f o r all δ ∈ Γ if and only if R is a crossed pro duct. F or a graded ring R and δ , α ∈ Γ, it is clear that R ( α ) ∼ = gr R ( δ ) as graded R -mo dules if and o nly if R ( α )( δ − 1 ) ∼ = gr R ( δ )( δ − 1 ); that is, if and only if R ( δ − 1 α ) ∼ = gr R . Then b y Prop osition 4.2.1 , R ( α ) ∼ = gr R ( δ ) as graded R -mo dules if a nd only if δ − 1 α ∈ Γ ∗ R . Chapter 4. G r a ded Azuma y a Alg ebras 71 F or a Γ-graded ring R , consider M n × m ( R ), the set of n × m matrices o ve r R . F or n = m , in Section 4 .4 , w e will deﬁne grading on the matrix ring M n ( R ), whic h giv es M n ( R ) the structure of a graded ring. Here w e deﬁne shifted matrices M n × m ( R ), whic h will b e used in the followin g prop osition. Note that they do not ha v e the structure o f either graded rings or graded R -mo dules. F or ( d ) = ( δ 1 , . . . , δ n ) ∈ Γ n , ( a ) = ( α 1 , . . . α m ) ∈ Γ m , let M n × m ( R )[ d ][ a ] =         R δ − 1 1 α 1 R δ − 1 1 α 2 · · · R δ − 1 1 α m R δ − 1 2 α 1 R δ − 1 2 α 2 · · · R δ − 1 2 α m . . . . . . . . . . . . R δ − 1 n α 1 R δ − 1 n α 2 · · · R δ − 1 n α m         So M n × m ( R )[ d ][ a ] consists of matrices with the ij -en try in R δ − 1 i α j . F urther, w e let GL n × m ( R )[ d ][ a ] denote the set of in v ertible n × m matrices with shifting as a b o ve . The follo wing prop osition extends Prop osition 4.2.1 . A similar ar gumen t will b e used in the pro of of Prop osition 5.4.3 . Prop osition 4.2.2. L et R b e a Γ -gr ade d ring and let ( d ) = ( δ 1 , . . . , δ n ) ∈ Γ n , ( a ) = ( α 1 , . . . , α m ) ∈ Γ m . Then R n ( d ) ∼ = gr R m ( a ) as gr ade d R -mo d ules if and only if ther e exists ( r ij ) ∈ GL n × m ( R )[ d ][ a ] . Pro of. If r = ( r ij ) ∈ GL n × m ( R )[ d ][ a ], then there is a graded R -mo dule homomor- phism R r : R n ( d ) − → R m ( a ) ( x 1 , . . . , x n ) 7− → ( x 1 , . . . , x n ) r . Since r is in v ertible, there is a matrix t ∈ GL m × n ( R ) with r t = I n and tr = I m . So there is an R -mo dule homomorphism R t : R m ( a ) → R n ( d ), whic h is an inv erse of R r . This pro v es tha t R r is bijectiv e, and therefore it is a graded R -mo dule isomorphism. Con v ersely , if φ : R n ( d ) ∼ = gr R m ( a ), then w e can construct a mat r ix as follo ws. Let e i denote the basis elemen t of R n ( d ) with 1 in the i -th en try and zeros els ewhere. Chapter 4. G r a ded Azuma y a Alg ebras 72 Then le t φ ( e i ) = ( r i 1 , r i 2 , . . . , r im ), and let r = ( r ij ) n × m . It can b e easily v eriﬁed that r ∈ M n × m ( R )[ d ][ a ]. In the same w ay , using φ − 1 : R m ( a ) → R n ( d ) construct a matrix t . Let e ′ i denote the i -th elemen t of the standard basis f or R m ( a ). Since e i = φ − 1 ◦ φ ( e i ) = r i 1 φ − 1 ( e ′ 1 ) + r i 2 φ − 1 ( e ′ 2 ) + · · · + r im φ − 1 ( e ′ m ) for eac h i , and in a similar wa y f o r φ ◦ φ − 1 , w e can sho w that r t = I n and tr = I m . So ( r ij ) ∈ GL n × m ( R )[ d ][ a ].  F or con v enience, in the ab o ve deﬁnition of M n × m ( R )[ d ][ a ], if ( a ) = ( e, . . . , e ), w e will write M n × m ( R )[ d ] instead of M n × m ( R )[ d ][ e ]. W e let Γ ∗ M n × m ( R ) =  ( d ) ∈ Γ n : GL n × m ( R )[ d ] 6 = ∅  . Then it is immediate from the ab ov e prop osition that R n ( d ) ∼ = gr R n as graded R - mo dules if and only if ( d ) ∈ Γ ∗ M n ( R ) . A Γ-g r a ded ring D = L γ ∈ Γ D γ is called a gr ade d divis i o n ring if ev ery non- zero ho mogeneous elemen t has a multiplicativ e in v erse. It follows from Prop osi- tion 4.1.2 (4) that Γ D is a gro up, so w e can write D = L γ ∈ Γ D D γ . Then, as a Γ D -graded ring, D is a crossed pro duct and it follo ws fro m Propo sition 4.1.3 (3) that D is strongly Γ D -graded. Examples 4.2.3. 1. Let E b e a division ring and let D = E [ x, x − 1 ] b e the La u- ren t p olynomials ov er E . Then D is a gra ded division ring, with D = L n ∈ Z D n where D n = { ax n : a ∈ E } . 2. Let H = R ⊕ R i ⊕ R j ⊕ R k b e the real quaternion alg ebra, with m ultiplication deﬁned by i 2 = − 1, j 2 = − 1 and ij = − j i = k . It is kno wn that H is a non-comm utative division r ing with cen tre R . W e note that H can b e giv en t w o diﬀeren t graded divis ion ring s tructures, with grade groups Z 2 and Z 2 × Z 2 resp ectiv ely . F or Z 2 : Let C = R ⊕ R i . Then H = C 0 ⊕ C 1 , where C 0 = C and C 1 = C j . Chapter 4. G r a ded Azuma y a Alg ebras 73 F or Z 2 × Z 2 : Let H = R (0 , 0) ⊕ R (1 , 0) ⊕ R (0 , 1) ⊕ R (1 , 1) , where R (0 , 0) = R , R (1 , 0) = R i, R (0 , 1) = R j, R (1 , 1) = R k . In b oth cases, it is routine to sho w H tha t for ms a graded division ring. In the follo wing prop ositions w e are considering graded mo dules ov er graded division rings. W e note that the grade groups here are deﬁned as ab ov e; that is, w e do not initially assume them to b e ab elian or torsion-free. The pro of s fo llow the standard pro ofs in the non-g r aded setting (see [ 37 , § IV, Thms. 2.4, 2 .7 , 2.13]), or the graded setting (see [ 5 , Thm. 3], [ 5 1 , Prop. 4.6 .1], [ 39 , p. 79]); ho w ev er extra care needs to b e giv en since the g r a ding is neither ab elian nor torsion-free. Prop osition 4.2.4. L et Γ b e a gr oup which a c ts f r e el y on a set Γ ′ . L et R b e a Γ -gr ade d di v ision rin g and M b e a Γ ′ -gr ade d mo dule over R . T hen M is a gr ade d fr e e R -m o dule. Mor e gener al ly, any line arly indep endent subset of M c onsis ting of homo ge n e ous elem ents c an b e extende d to form a homo gene ous b asi s of M . Pro of. Note that the ﬁrst statemen t is an immediate consequence of the second, since for any m ∈ M h , { m } is a linearly independen t subset of M . Fix a linearly indep enden t subset X o f M consisting of homogeneous elemen ts. Let F = { Q ⊆ M h : X ⊆ Q and Q is R -linearly indep enden t } . This is a non-empty partially ordered set with inclusion, and ev ery c hain Q 1 ⊆ Q 2 ⊆ . . . in F has an upp er b ound S Q i ∈ F . By Zo rn’s Lemma, F has a maximal elemen t, whic h we denote by P . If h P i 6 = M , then there is a ho mogeneous elemen t m ∈ M h \ h P i . W e will sho w that P ∪ { m } is a linearly indep enden t set con taining X , con tradicting our c hoice of P . Supp ose r m + P r i p i = 0, where r, r i ∈ R , p i ∈ P and r 6 = 0. Since r 6 = 0 there is a homog eneous comp onen t of r , sa y r λ , whic h is also non- zero. Conside ring the λ deg( m )-homogeneous comp o nen t of this sum, w e ha v e m = r − 1 λ r λ m = − P r − 1 λ r i p i Chapter 4. G r a ded Azuma y a Alg ebras 74 for r i homogeneous, which contradicts our c hoice of m . Hence r = 0, whic h implies eac h r i = 0. This giv es the required con tradiction, so M = h P i , completing the pro of.  Prop osition 4.2.5. L et Γ b e a gr oup which a c ts f r e el y on a set Γ ′ . L et R b e a Γ -gr ade d division ring and M b e a Γ ′ -gr ade d mo dule over R . T h en any two homo ge- ne ous b ases of M over R have the sam e c ar dina lity. Pro of. Supp ose M has an inﬁnite basis Z . Since R is in particular a ring, w e can apply [ 37 , Thm. IV .2.6 ], whic h sho ws that ev ery basis of M has the same cardinalit y as Z . W e now assume that M has t w o homogeneous bases X and Y , where X and Y are ﬁnite. Then X = { x 1 , . . . , x n } and Y = { y 1 , . . . , y m } for x i , y i ∈ M h \ 0. As X is a basis f or M , w e can write y m = r 1 x 1 + · · · + r n x n for some r i ∈ R h , where deg ( y m ) = deg( r i ) deg( x i ) for eac h i . Since y m 6 = 0, w e hav e at least one r i 6 = 0. Let r k b e the ﬁrst non-zero r i , and we no te that r k is in ve rtible as it is non-zero and homogeneous in R . Then x k = r − 1 k y m − r − 1 k r k +1 x k +1 − · · · − r − 1 k r n x n , and the set X ′ = { y m , x 1 , . . . , x k − 1 , x k +1 , . . . , x n } spans M since X spans M , where X ′ consists of homogeneous elemen ts. So y m − 1 = s m y m + t 1 x 1 + · · · + t k − 1 x k − 1 + t k +1 x k +1 + · · · + t n x n for s m , t i ∈ R h . There is at least one no n-zero t i , since if all t he t i are zero, t hen either y m and y m − 1 are linearly dep endent or y m − 1 is zero. Let t j denote the ﬁrst non-zero t i . Then x j can b e written as a linear com bination of y m − 1 , y m and those x i with i 6 = j, k . Therefore the set X ′′ = { y m − 1 , y m } ∪ { x i : i 6 = j, k } spans M since X ′ spans M . W e can write y m − 2 as a linear com bination of the elemen ts of X ′′ . Chapter 4. G r a ded Azuma y a Alg ebras 75 Con tin uing this pro cess of adding a y and r emoving an x giv es, after the k -th step, a set whic h spans M consisting of y m , y m − 1 , . . . , y m − k + 1 and n − k of the x i . If n < m , then a f ter the n -th step, w e w o uld hav e that the set { y m , . . . , y m − n +1 } spans M . But if n < m , then m − n + 1 ≥ 2, so this set do es not contain y 1 , and therefore y 1 can b e written as a linear com bination of the elemen ts of this set. This con tradicts the linear indep endence of Y , so w e m ust hav e m ≤ n . Rep eating a similar argumen t with X a nd Y in terc hanged gives n ≤ m , so n = m .  The prop o sitions ab ov e sa y that f o r a graded mo dule M o v er a graded division ring R , M has a homogeneous basis and an y t w o homogeneous ba ses of M ha v e the same cardinality . The cardinal n um b er of a ny homogeneous basis o f M is called the dimension of M ov er R , and it is denoted by dim R ( M ). Prop osition 4.2.6. L et Γ b e a gr oup which acts fr e ely on a set Γ ′ . L et R b e a Γ - gr ade d division ring and M b e a Γ ′ -gr ade d mo dule over R . If N is a gr ade d submo d ule of M , then dim R ( N ) + dim R ( M / N ) = dim R ( M ) . Pro of. Let Y b e a homogeneous basis of N . Then Y is a linearly indep enden t subset of M consisting of homogeneous elemen ts, so using Prop osition 4.2.4 , there is a homogeneous ba sis X of M con taining Y . W e will show that U = { x + N : x ∈ X \ Y } is a homogeneous basis of M / N . Note that clearly U consists of homogeneous elemen ts. Let m + N ∈ ( M / N ) h . Then m ∈ M h and m = P r i x i + P s j y j where r i , s j ∈ R , y j ∈ Y and x i ∈ X \ Y . So m + N = P r i ( x i + N ), whic h sho ws that U spans M / N . If P r i ( x i + N ) = 0, for r i ∈ R , x i ∈ X \ Y , then P r i x i ∈ N whic h implies that P r i x i = P s k y k for s k ∈ R and y k ∈ Y . But X = Y ∪ ( X \ Y ) is linearly indep enden t, so r i = 0 and s k = 0 for all i , k . Therefore U is a homogeneous basis for M / N and as w e can construct a bijectiv e map U → X \ Y , w e hav e | U | = | X \ Y | . Then dim R M = | X | = | Y | + | X \ Y | = | Y | + | U | = dim R N + dim R ( M / N ).  Chapter 4. G r a ded Azuma y a Alg ebras 76 4.3 Graded cen tr al simple algeb ras Let Γ b e a multiplicativ e gro up and let R b e a comm utativ e Γ-gra ded ring. A Γ -gr ade d R -algebr a A = L γ ∈ Γ A γ is deﬁned to b e a graded ring whic h is an R - algebra suc h that the asso ciated ring homomorphism ϕ is a graded homomorphism, where ϕ : R → A with ϕ ( R ) ⊆ Z ( A ). Graded R -alg ebra homomorphisms and graded subalgebras a re deﬁned analog ously to the equiv alen t terms f o r graded ring s or graded mo dules. Let A and B b e graded R -algebras, suc h tha t Γ A ⊆ Z Γ (Γ B ), where Z Γ (Γ B ) is t he set of elemen ts of Γ whic h comm ute with Γ B . Then A ⊗ R B has a nat ural grading as a graded R -algebra give n by A ⊗ R B = L γ ∈ Γ ( A ⊗ R B ) γ where: ( A ⊗ R B ) γ = ( X i a i ⊗ b i : a i ∈ A h , b i ∈ B h , deg( a i ) deg( b i ) = γ ) Note that the condition Γ A ⊆ Z Γ (Γ B ) is needed to ensure that the m ultiplication on A ⊗ R B is w ell deﬁned. Let R b e a Γ -graded ring, M b e a Γ- graded R -mo dule and let END R ( M ) = HOM R ( M , M ), where HOM R ( M , M ) is deﬁned on page 69 . The ring E ND R ( M ) is a Γ-graded r ing with the usual addition and with m ultiplicatio n deﬁned as g · f = f ◦ g for all f , g ∈ END R ( M ). If M is graded free with a ﬁnite homogeneous base, then we ha v e END R ( M ) = End R ( M ) (see [ 51 , Remark 2.10.6(ii)]). In fa ct, if M and N ar e Γ-graded R -mo dules with M ﬁnitely generated, then HOM R ( M , N ) = Hom R ( M , N ) (see [ 51 , Cor. 2 .4.4]). F urther, if R is a commutativ e Γ-graded ring and Γ R ⊆ Z Γ (Γ M ), then End R ( M ) is a Γ-graded R -algebra. A g r ade d ﬁeld is deﬁned to b e a comm utativ e g raded division ring. Note that the supp ort of a graded ﬁeld is an ab elian group. Let R b e a graded ﬁeld. A graded algebra A ov er R is said to b e a gr ade d c entr a l s i m ple al g e br a o v er R if A is graded simple a s a graded ring, Z ( A ) ∼ = gr R and [ A : R ] < ∞ . Note that since the cen tre of A is R , whic h is a graded ﬁeld, A is graded free as a g r aded mo dule ov er its cen tre b y Prop osition 4.2.4 , so the dimension of A o ve r R is uniquely deﬁned. Chapter 4. G r a ded Azuma y a Alg ebras 77 F or a Γ- graded ring A , let A op denote the opp osite g raded ring, where the grade group of A op is the o pp osite group Γ op . So for a graded R -algebra A , in order to deﬁne A ⊗ R A op , w e note that the supp ort o f A m ust b e ab elian. Moreo v er for the follo wing prop ositions, w e require that the group Γ is an ab elian gr o up (see Theorem 4.3.5 and the preceding commen ts). Sinc e the gr ad e gr oups ar e assume d to b e ab elian for the r emainder of this se ction, we wil l write them as ad ditive gr oups. By combin ing Prop ositions 4.3.1 and 4.3.2 , w e sho w that the tensor pro duct of t w o gra ded cen t r a l simple R -alg ebras is gra ded cen tr a l simple, w here the grade gro up Γ, as b elow, is abelian but not neces sarily torsion-free. This has been prov en b y W all for graded cen tral simple algebras with Z / 2 Z as the support (see [ 72 , Thm. 2]), a nd b y Hw ang and W adsw orth for R -algebras with a torsion-free grade group (see [ 39 , Prop. 1.1]). Prop osition 4.3.1. L et Γ b e an ab elian gr oup. L et R b e a Γ -gr ade d ﬁeld and let A and B b e Γ -gr ade d R -algebr as. If A is gr ade d c entr al simple o v er R and B is gr ade d simple, then A ⊗ R B i s g r ade d simp le. Pro of. Let I b e a homogeneous t w o-sided ideal of A ⊗ B , with I 6 = 0. W e will sho w that A ⊗ B = I . Note that since I is a homogeneous ideal, b y Prop osition 4.1.4 , it is generated b y homog eneous elemen t s. First supp o se a ⊗ b is a homogeneous elemen t of I , where a ∈ A h and b ∈ B h . Then A is the homogeneous tw o-sided ideal generated b y a , so there exist a i , a ′ i ∈ A h with 1 = P a i aa ′ i . Then X ( a i ⊗ 1)( a ⊗ b )( a ′ i ⊗ 1) = 1 ⊗ b is an elemen t of I . Similarly , B is t he homogeneous tw o-sided ideal generated b y b . Rep eating the ab ov e arg umen t sho ws that 1 ⊗ 1 is an elemen t of I , provin g I = A ⊗ B in this case. No w supp ose there is a n elemen t x ∈ I h , where x = a 1 ⊗ b 1 + · · · + a k ⊗ b k , with a j ∈ A h , b j ∈ B h and k as small as po ssible. No t e that since x is homogeneous, deg( a j ) + deg( b j ) = deg ( x ) for a ll j . By the ab o v e a rgumen t w e can supp ose that Chapter 4. G r a ded Azuma y a Alg ebras 78 k > 1. As ab o v e, since a k ∈ A h , there are c i , c ′ i ∈ A h with 1 = P c i a k c ′ i . Then X ( c i ⊗ 1) x ( c ′ i ⊗ 1) =  X ( c i a 1 c ′ i )  ⊗ b 1 + · · · +  X ( c i a k − 1 c ′ i )  ⊗ b k − 1 + 1 ⊗ b k , where the terms ( P i ( c i a j c ′ i )) ⊗ b j are homogeneous elemen ts of A ⊗ B . Th us, without loss of generality , w e can a ssume that a k = 1. Then a k and a k − 1 are linearly indep en- den t, since if a k − 1 = ra k with r ∈ R , then a k − 1 ⊗ b k − 1 + a k ⊗ b k = a k ⊗ ( r b k − 1 + b k ), whic h is homogeneous and thus gives a smaller v alue of k . Th us a k − 1 / ∈ R = Z ( A ), and so there is a homogeneous elemen t a ∈ A with aa k − 1 − a k − 1 a 6 = 0. Consider the comm utator ( a ⊗ 1) x − x ( a ⊗ 1) = ( aa 1 − a 1 a ) ⊗ b 1 + · · · + ( aa k − 1 − a k − 1 a ) ⊗ b k − 1 , where the last summand is not zero. If the whole sum is not zero, then we hav e constructed a homogeneous elemen t in I with a smaller k . Otherwise supp ose the whole sum is zero, and write c = aa k − 1 − a k − 1 a . Then w e can write c ⊗ b k − 1 = P k − 2 j =1 x j ⊗ b j where x j = − ( aa j − a j a ). Since 0 6 = c ∈ A h and A is t he homog eneous t w o-sided ideal generated b y c , using the same argument as ab o ve , w e ha ve 1 ⊗ b k − 1 = x ′ 1 ⊗ b 1 + · · · + x ′ k − 2 ⊗ b k − 2 (4.1) for some x ′ j ∈ A h . Since b 1 , . . . , b k − 1 are linearly indep enden t homogeneous ele- men ts of B , they can b e extended to form a homo g eneous basis of B , say { b i } , by Prop osition 4.2.4 . Then { 1 ⊗ b i } forms a homogeneous basis of A ⊗ R B as a graded A -mo dule, so in particular they are A -linearly indep enden t, whic h is a contradiction to equation ( 4.1 ). This reduces the pro of to the ﬁrst case.  Prop osition 4.3.2. L et Γ b e an ab elian gr oup. L et R b e a Γ -gr ade d ﬁeld and let A and B b e Γ -g r ade d R -algebr a s . If A ′ ⊆ A and B ′ ⊆ B ar e gr ade d sub a l g e br as, then Z A ⊗ B ( A ′ ⊗ B ′ ) = Z A ( A ′ ) ⊗ Z B ( B ′ ) . Chapter 4. G r a ded Azuma y a Alg ebras 79 In p articular, if A a nd B ar e c entr al over R , then A ⊗ R B i s c entr al. Pro of. First note that b y Prop osition 4.2.4 , A ′ , B ′ , Z A ( A ′ ) and Z B ( B ′ ) are graded free ov er R , so they are ﬂat o v er R . Th us w e can consider Z A ⊗ B ( A ′ ⊗ B ′ ) and Z A ( A ′ ) ⊗ Z B ( B ′ ) as graded subalgebras of A ⊗ B . The inclusion ⊇ fo llo ws immediately . F or the rev erse inc lusion, le t x ∈ Z A ⊗ B ( A ′ ⊗ B ′ ). Let { b 1 , . . . , b n } b e a homogeneous basis for B ov er R whic h exists thanks to Prop osition 4.2.4 . Then x can b e written uniquely as x = x 1 ⊗ b 1 + · · · + x n ⊗ b n for x i ∈ A ( see [ 37 , Thm. IV.5.11]). F or ev ery a ∈ A ′ , ( a ⊗ 1) x = x ( a ⊗ 1), so ( ax 1 ) ⊗ b 1 + · · · + ( ax n ) ⊗ b n = ( x 1 a ) ⊗ b 1 + · · · + ( x n a ) ⊗ b n . By the uniqueness of this represen tation w e hav e x i a = ax i , so that x i ∈ Z A ( A ′ ) for eac h i . Th us we ha ve sho wn that x ∈ Z A ( A ′ ) ⊗ R B . Similarly , let { c 1 , . . . , c k } b e a ho- mogeneous basis of Z A ( A ′ ). Then w e can write x uniquely as x = c 1 ⊗ y 1 + · · · + c k ⊗ y k for y i ∈ B . A similar argumen t to ab ov e show s that y i ∈ Z B ( B ′ ), completing the pro of.  In the following theorem, since Γ is an ab elian group, we deﬁne m ultiplication in End R ( A ) to b e g · f = g ◦ f . Theorem 4.3.3. L et Γ b e an ab elian gr oup. L e t A b e a Γ -gr ade d c entr a l simple algebr a over a Γ -gr ade d ﬁ eld R . Th e n A is an Azumaya algebr a over R . Pro of. Since A is graded free of ﬁnite dimension ov er R , it follows that A is faithfully pro jectiv e ov er R . There is a natural g r a ded R -algebra homomorphism ψ : A ⊗ R A op → End R ( A ) deﬁned b y ψ ( a ⊗ b )( x ) = axb where a, x ∈ A , b ∈ A op . Since graded ideals of A op are the same as graded ideals o f A , w e ha ve that A op is gr a ded simple. By Prop osition 4.3.1 , A ⊗ A op is also graded simple, so ψ is injectiv e. He nce the map is surjectiv e b y dimension count, using Theorem 4.2.6 . This shows that A is a n Azuma y a algebra o v er R , as required.  Chapter 4. G r a ded Azuma y a Alg ebras 80 Corollary 4.3.4. L et Γ b e an ab elian gr oup. L et A b e a Γ -gr ade d c entr al simple algebr a over a Γ -gr ade d ﬁ eld R of dimension n . Then for any i ≥ 0 , K i ( A ) ⊗ Z [1 /n ] ∼ = K i ( R ) ⊗ Z [1 /n ] . Pro of. By Theorem 4.3.3 , a gra ded cen t r a l simple algebra A ov er R is an Azumay a algebra. F ro m Prop osition 4.2.4 , since R is a graded ﬁeld, A is a free R - mo dule. The corollary now f ollo ws immediately fr o m Theorem 3.1.5 , since A is an Azuma y a algebra free o v er its cen tr e.  F or a graded ﬁeld R , Theorem 4.3.3 sho ws that a graded cen tra l simple R -algebra, graded b y an ab elian group Γ, is an Azuma ya algebra o v er R . This theorem can not be extende d to co ver non- a b elian grading. Consider a ﬁnite dimen sional division algebra D and a group G . The group ring D G is a graded division ring, so is clearly a graded simple algebra, and if G is a b elian, Theorem 4.3.3 implies that D G is an Azuma y a alg ebra. Ho w ev er in general, for an arbitra r y gro up G , D G is no t alw a ys an Azuma ya algebra. DeMey er a nd Jan usz [ 18 ] ha v e pr ov en the follow ing theorem. Theorem 4.3.5. L et R b e a ri n g and le t G b e a gr oup. Then the g r oup ring RG is an Azumaya algeb r a if and only if the fol lowing thr e e c onditions hold: 1. R is an Azumaya algebr a, 2. [ G : Z ( G )] < ∞ , 3. the c ommutator sub gr o up of G has ﬁnite or der m and m is invertible in R . Pro of. See [ 18 , Thm. 1].  4.4 Graded matrix ring s Let Γ b e a group and R b e a Γ-graded ring. W e will write Γ as a m ultiplicativ e group, since Γ is not necessarily ab elian. Througho ut this section, unless otherwise Chapter 4. G r a ded Azuma y a Alg ebras 81 stated, w e will a ssume that a ll graded r ing s, graded mo dules and graded algebras are also Γ-graded. Let λ ∈ Γ and ( d ) = ( δ 1 , . . . , δ n ) ∈ Γ n . Let M n ( R )( d ) λ denote the n × n - matrices o v er R with the degree shifted as follows : M n ( R )( d ) λ =         R δ 1 λδ − 1 1 R δ 1 λδ − 1 2 · · · R δ 1 λδ − 1 n R δ 2 λδ − 1 1 R δ 2 λδ − 1 2 · · · R δ 2 λδ − 1 n . . . . . . . . . . . . R δ n λδ − 1 1 R δ n λδ − 1 2 · · · R δ n λδ − 1 n         Th us M n ( R )( d ) λ consists of matrices with the ij -entry in R δ i λδ − 1 j . Prop osition 4.4.1. With the notation as ab ove , ther e is a gr a d e d ring M n ( R )( d ) = M λ ∈ Γ M n ( R )( d ) λ . Pro of. (See [ 51 , Prop. 2.10.4].) F or any λ 1 , λ 2 ∈ Γ, M n ( R )( d ) λ 1 M n ( R )( d ) λ 2 ⊆ M n ( R )( d ) λ 1 λ 2 . F or an y i, j with 1 ≤ i, j ≤ n , R δ i λδ − 1 j ∩ P γ 6 = λ R δ i γ δ − 1 j = 0, so M n ( R )( d ) λ ∩ X γ 6 = λ M n ( R )( d ) γ ! = 0 . An y matrix in M n ( R ) can b e written as a sum o f matrices with a homog eneous elemen t in one en try and zeros elsewhere. If A ∈ M n ( R ) has a ∈ R ǫ in the ij -en try and zeros elsewhere, then taking λ = δ − 1 i ǫ δ j giv es that A ∈ M n ( R )( d ) λ , and hence an y matrix in M n ( R ) can b e written as an elemen t of M n ( R )( d ).  Let M b e a graded R -mo dule which is graded f ree with a ﬁnite homogeneous base { b 1 , . . . , b n } , where deg ( b i ) = δ i . W e no t ed in the pr evious section that the ring End R ( M ) is a graded ring with m ultiplication deﬁned as g · f = f ◦ g for all f , g ∈ End R ( M ). If w e igno re the grading, it is w ell-know n that End R ( M ) ∼ = M n ( R ) Chapter 4. G r a ded Azuma y a Alg ebras 82 as rings (see [ 37 , Thm . VI I.1.2]). When w e ta k e the grading in to accoun t, the follo wing prop osition sho ws that this isomorphism is in fa ct a graded isomorphism. Prop osition 4.4.2. L et M b e a gr ade d f r e e Γ -gr ade d R -mo dule with a ﬁnite ho- mo gene ous b ase { b 1 , . . . , b n } , wher e deg ( b i ) = δ i . Then End R ( M ) ∼ = gr M n ( R )( d ) as Γ -gr ade d rings, whe r e ( d ) = ( δ 1 , . . . , δ n ) ∈ Γ n . Pro of. (See [ 51 , Prop. 2.1 0.5].) The remarks b efore the pro p osition sho w that End R ( M ) ∼ = M n ( R ). T o sho w that the isomorphism is graded, let f ∈ End R ( M ) λ for some λ ∈ Γ. Then as in the non-graded setting, there are elemen t s r ij ∈ R suc h that f ( b 1 ) = r 11 b 1 + r 12 b 2 + · · · + r 1 n b n f ( b 2 ) = r 21 b 1 + r 22 b 2 + · · · + r 2 n b n . . . f ( b n ) = r n 1 b 1 + r n 2 b 2 + · · · + r nn b n , with associat ed matrix ( r ij ) ∈ M n ( R ). Since w e ha v e f ( b i ) ∈ M δ i λ for each b i , it follo ws that eac h r ij is homo g eneous of degree δ i λδ − 1 j . So ( r ij ) ∈ M n ( R )( d ) λ as required.  Remark 4.4.3. Note that ab ov e all graded R -mo dules are considered as left mo d- ules. F or the λ -comp o nen t of the graded matrix ring M n ( R )( d ), we s et the degree of the ij -en try to b e δ i λδ − 1 j . Then in the ab ov e prop o sition, w e deﬁned the m ultipli- cation in End R ( M ) as g · f = f ◦ g to ensure tha t the isomorphism is a gra ded ring isomorphism. If w e deﬁne multiplic atio n in End R ( M ) by g · f = g ◦ f , then in the non-g r a ded setting, we hav e a ring isomorphism End R ( M ) → M n ( R op ), where this map is the comp osition of the homomorphism in the ab ov e prop osition with the tra nsp ose map. T o mak e End R ( M ), with this multiplication, into a graded ring, it will b e graded b y Γ op . In the ab o v e prop osition, we will ha v e End R ( M ) ∼ = gr M n ( R op )( d ), where Chapter 4. G r a ded Azuma y a Alg ebras 83 M n ( R op )( d ) is a lso Γ op -graded and the degree of the ij -en try of M n ( R op )( d ) is deﬁned to b e δ − 1 i · op λ · op δ j . In D eﬁnition 4.5.3 and in Section 5.4 , since Γ is an ab elian g roup and R is a comm utativ e graded ring, w e will deﬁne m ultiplication in endomorphism rings to b e g · f = g ◦ f . Then w e will use the gra ding on matrix rings men tioned in the previous paragraph. Supp ose M is a graded righ t R -mo dule and m ultiplication in End R ( M ) is deﬁned b y g · f = g ◦ f (see page 85 for some commen ts on graded right R -mo dules). Then to get a graded ring isomorphism End R ( M ) ∼ = gr M n ( R )( d ), w e need to deﬁne the grading on the matrix ring M n ( R )( d ) as having its ij -en try in the λ -comp onen t of degree δ − 1 i λδ j . With a g raded R -mo dule M deﬁned as in the ab ov e prop osition, we deﬁne the graded R - mo dule homomor phisms E ij b y E ij ( b l ) = δ il b j for 1 ≤ i, j, l ≤ n , where δ il is the Kronec k er delta. W e note that the graded R -mo dule homomorphisms E ij , for 1 ≤ i, j, l ≤ n , are ho mo g eneous elemen ts in End R ( M ) of degree δ − 1 i δ j . Then { E ij : 1 ≤ i, j ≤ n } forms a homog eneous basis for End R ( M ). Via the isomorphism End R ( M ) ∼ = gr M n ( R )( d ), the map E ij corresp onds to the matrix e ij , where e ij is the matrix with 1 in the ij -en try and zeros elsewhe re. This o bserv ation will b e used in the fo llo wing prop osition to prov e some basic prop erties of the graded matrix ring s. Prop osition 4.4.4. L et R b e a Γ -gr ade d ring and le t ( d ) = ( δ 1 , . . . , δ n ) ∈ Γ n . 1. If π ∈ S n is a p ermutation, then M n ( R )( δ 1 , . . . , δ n ) ∼ = gr M n ( R )( δ π (1) , . . . , δ π ( n ) ) . 2. If ( γ 1 , . . . , γ n ) ∈ Γ n with e ach γ i ∈ Γ ∗ R , then M n ( R )( δ 1 , . . . , δ n ) ∼ = gr M n ( R )( γ 1 δ 1 , . . . , γ n δ n ) . If R is a g r ade d division ring, then any se t of ( γ 1 , . . . , γ n ) ∈ Γ n R c an b e chosen . Chapter 4. G r a ded Azuma y a Alg ebras 84 3. If σ ∈ Z (Γ ) , the c entr e of Γ , then M n ( R )( δ 1 , . . . , δ n ) = M n ( R )( δ 1 σ , . . . , δ n σ ) . Pro of. (See [ 39 , p. 78], [ 51 , Remarks 2.10.6].) W e observ ed on page 69 that M = R ( δ − 1 1 ) ⊕ · · · ⊕ R ( δ − 1 n ) has a homogeneous basis { e 1 , . . . , e n } with deg( e i ) = δ i . So fro m the ab o ve pro p o- sition, w e hav e End R ( M ) ∼ = gr M n ( R )( δ 1 , . . . , δ n ), where the map E ij corresp onds to the matrix e ij . (1) Let δ π ( i ) = τ i . Then N = R ( τ − 1 1 ) ⊕ · · · ⊕ R ( τ − 1 n ) has a standard homogeneous basis { e ′ 1 , . . . , e ′ n } with deg( e ′ i ) = τ i , and w e hav e End R ( N ) ∼ = gr M n ( R )( τ 1 , . . . , τ n ) . W e deﬁne a g raded R -mo dule isomorphism φ : M → N b y φ ( e i ) = e ′ π − 1 ( i ) , whic h induces a gra ded isomorphism φ : End R ( M ) → End R ( N ); f 7→ φ ◦ f ◦ φ − 1 . Combining these graded isomorphisms giv es the required result: M n ( R )( δ 1 , . . . , δ n ) ∼ = gr End A ( M ) φ − → End R ( N ) ∼ = gr M n ( R )( δ π (1) , . . . , δ π ( n ) ) . (2) Let u 1 , . . . , u n ∈ R b e homogeneous units of R with deg( u i ) = γ i . Let N = R ( δ − 1 1 ) ⊕ · · · ⊕ R ( δ − 1 n ) , w here w e are cons idering N with t he homogeneous basis { u 1 e 1 , . . . , u n e n } with deg ( u i e i ) = γ i δ i . Then with this basis, End R ( N ) ∼ = gr M n ( R )( γ 1 δ 1 , . . . , γ n δ n ) . Deﬁne a graded R -mo dule isomorphism φ : M → N by φ ( e i ) = u − 1 i ( u i e i ). The required isomorphism follows in a similar wa y to (1). C learly , if R is a graded division ring then ev ery non-zero homog eneous elemen t is inv ertible, so any set of Chapter 4. G r a ded Azuma y a Alg ebras 85 ( γ 1 , . . . , γ n ) ∈ Γ n R can b e c hosen here. (3) Since σ ∈ Z (Γ), it is clear tha t M n ( R )( δ 1 , . . . , δ n ) λ = M n ( R )( δ 1 σ , . . . , δ n σ ) λ for eac h λ ∈ Γ, as required.  W e no t e that if R is a Γ-g raded ring, then for a Γ-graded righ t R -mo dule M , w e ha v e M λ R γ ⊆ M λγ for all λ, γ ∈ Γ. Then as for g raded left R - mo dule homomor- phisms, g raded righ t R -mo dule ho mo mo r phisms b et w een g raded rig h t R - mo dules ma y shift the gra ding . W e note that they a r e left shifted; that is, if M , N are graded right R -mo dules and f ∈ Hom R ( M , N ) δ , t hen f ( M γ ) ⊆ N δγ for all γ ∈ Γ. The δ -shifted right R -mo dule ( δ ) M is deﬁned as ( δ ) M = L γ ∈ Γ  ( δ ) M  γ where  ( δ ) M  γ = M δγ . The follow ing four results (Prop osition 4.4.5 to Corollary 4.4.8 ) are the graded v ersions of some results on simple ring s (see [ 37 , § IX.1]). These are required for the pro of of Theorem 4.4.9 . Prop osition 4.4.5. L e t D b e a gr a d e d division ring, let V b e a ﬁnite di m ensional gr ade d mo dule over D , and le t R = End D ( V ) . If A and B ar e gr a de d right R -mo dules which a r e faithful and gr ade d simple, then ( γ ) A ∼ = gr B as gr ade d ri g ht R -mo d ules for some γ ∈ Γ . Pro of. F ro m Prop osition 4.4.2 , R ∼ = gr M n ( D )( d ) for some ( d ) ∈ Γ n . Then w e will sho w that M n ( D )( d ) contains a minimal graded righ t ideal. F or some i , consider J 1 = { e i,i X : X ∈ M n ( D )( d ) } , where e i,i is the elemen tary matr ix with 1 in the i, i -en t r y . Then J 1 consists of all matrices in M n ( D )( d ) with j -th row zero f or j 6 = i and J 1 is a graded right ideal of M n ( D )( d ). If J 1 is not minimal, there is a non-zero graded righ t ideal J 2 of M n ( D )( d ), with J 2 ⊆ J 1 . Since J 2 is a gra ded ideal, it con tains a non-zero homogeneous elemen t. Then by using the elemen tary matrices, w e can sho w that J 1 = J 2 . So R contains a minimal graded right ideal, whic h w e denote b y I . Since A is fait hf ul, its annihilator is zero, so there is a homogeneous elemen t a ∈ A ε for some ε ∈ Γ, suc h t hat aI 6 = 0 (if not, then aI = 0 for all a ∈ A h , so Chapter 4. G r a ded Azuma y a Alg ebras 86 I ⊆ Ann( A ) = 0 , con tradicting I 6 = 0). Then aI is a graded submo dule of A as it is generated b y the homogeneous elemen ts a ∈ A ε and all the i ∈ I h . But A is gra ded simple, so aI = A . Deﬁne a map ψ : I − → ( ε ) aI = ( ε ) A i 7− → ai. This is a graded righ t R -mo dule homomorphism, whic h is surjectiv e. Since k er( ψ ) is a graded right ideal of I , a nd I is minimal, t his implies the ke rnel is zero, so ψ is a graded isomorphism. Similarly , w e hav e I ∼ = gr ( ε ′ ) B f o r some ε ′ ∈ Γ. So ( ε ) A ∼ = gr ( ε ′ ) B , whic h says ( γ ) A ∼ = gr B fo r some γ ∈ Γ.  Prop osition 4.4.6. L et D b e a gr ade d d i v ision ring, V b e a non-zer o gr ade d D - mo dule, and let R = End D ( V ) . If g : V → V is a left shifte d homomorphis m of additive gr oups such that g f = f g for al l f ∈ R , then ther e is a h omo gen e ous element d ∈ D h such that g ( v ) = dv for al l v ∈ V . Pro of. Let u ∈ V h \ 0. W e will sho w u and g ( u ) are linearly dep enden t o ve r D . This is clear if dim D ( V ) = 1. Supp o se dim D ( V ) ≥ 2, and supp ose they a re linearly indep enden t. Then { u, g ( u ) } can b e extended to form a homogeneous basis of V . W e can deﬁne a map f ∈ End D ( V ) suc h that f ( u ) = 0 and f ( g ( u )) = v 6 = 0 for some v ∈ V . W e assumed f g = g f , so f ( g ( u )) = g ( f ( u )) = g (0) = 0, contradicting the c hoice of f ( g ( u )). So u and g ( u ) are linearly dep enden t ov er D , and t here is some d ∈ D h with g ( u ) = du . Le t v ∈ V , and let h ∈ R with h ( u ) = v . Then g ( v ) = g ( h ( u )) = h ( g ( u )) = h ( du ) = d ( h ( u )) = dv .  Prop osition 4.4.7. L et D , D ′ b e gr ade d division rin g s a nd let V , V ′ b e gr ade d mo dules of ﬁnite dimension n , n ′ over D , D ′ r esp e ctively. If End D ( V ) ∼ = gr End D ′ ( V ′ ) as gr ade d rin gs, then dim D ( V ) = dim D ′ ( V ′ ) and D ∼ = gr D ′ . Pro of. Let R = End D ( V ). Note that V is a gr aded righ t R -mo dule via v r = r ( v ) for all v ∈ V , r ∈ R . W e will sho w that V is fa ithful and graded simple as a graded Chapter 4. G r a ded Azuma y a Alg ebras 87 righ t R -mo dule. If V r = 0 for r ∈ R , then r ( v ) = 0 for all v ∈ V , so r = 0. T o sho w graded simple, let M b e a non-zero graded right R -submo dule of V , and let m ∈ M h \ 0. Extend { m } to form a homogeneous ba sis of V . F or some v ∈ V \ M , deﬁne a map θ v : V → V by θ v ( m ) = v and θ ( w ) = 0 for all other elemen ts w in this basis of V . Then θ v ∈ R , and mθ v = v / ∈ M , so M is not closed under action of R . This sho ws V is graded simple. Denote t he giv en graded ring isomorphism b y σ : R → End D ′ ( V ′ ). Then using the same argumen t as ab o v e, we ha v e that V ′ is a faithful and graded simple graded righ t End D ′ ( V ′ )-mo dule. Using the isomorphism σ , we can giv e V ′ the structure of a graded righ t R -mo dule by deﬁning w r = w σ ( r ) for eac h w ∈ V ′ , r ∈ R . It follows that V ′ is faithful with resp ect to R . An y graded r igh t R -submo dule of V ′ as also closed und er action of End D ′ ( V ′ ), so it is a graded righ t End D ′ ( V ′ )-submo dule of V ′ . This sho ws that V ′ is graded simple as a g raded righ t R -mo dule. Applying Prop osition 4.4.5 , there is a graded right R -mo dule isomorphism φ : ( γ ) V → V ′ for some γ ∈ Γ. Then for v ∈ V , f ∈ R , ( φ ◦ f )( v ) = φ ( f ( v )) = φ ( v f ) = φ ( v ) f = φ ( v ) σ ( f ) = σ ( f )  φ ( v )  , so that φ ◦ f ◦ φ − 1 = σ ( f ) as a homomorphism of additiv e gro ups from V ′ to V ′ . F or d ∈ D h , let α d : V → V ; x 7→ dx . This is a (left shifted) ho momorphism of additive groups. Clearly α d = 0 if and o nly if d = 0. Similarly fo r e ∈ D ′ h w e can deﬁne α e . Let f ∈ R , v ∈ V and d ∈ D h . Then ( f ◦ α d )( v ) = f ( d v ) = d f ( v ) = ( α d ◦ f )( v ) , so f ◦ α d = α d ◦ f . Then using the ab o v e results ( φ ◦ α d ◦ φ − 1 ) ◦ ( σ ( f )) = ( σ ( f )) ◦ ( φ ◦ α d ◦ φ − 1 ). Since σ is surjectiv e and φ ◦ α d ◦ φ − 1 is a left shifted homomorphism of additiv e groups from V ′ to V ′ , we can apply Prop osition 4 .4.6 . There is a homog eneous elemen t e ∈ D ′ h suc h that φ ◦ α d ◦ φ − 1 ( w ) = ew for all w ∈ V ′ . Chapter 4. G r a ded Azuma y a Alg ebras 88 Deﬁne a map τ : D h − → D ′ h d 7− → e and extend it linearly to cov er all of D . It is routine t o show that τ is a graded ring homomorphism. F or example, w e will sho w that, for some γ ∈ Γ, if d 1 , d 2 ∈ D γ then τ ( d 1 + d 2 ) = τ ( d 1 ) + τ ( d 2 ). The others parts are similar. Let e 1 , e 2 , e 3 ∈ D ′ h with φ ◦ α d 1 ◦ φ − 1 = α e 1 , φ ◦ α d 2 ◦ φ − 1 = α e 2 and φ ◦ α d 1 + d 2 ◦ φ − 1 = α e 3 . Then for w ∈ V ′ , e 3 w = φ ◦ α d 1 + d 2 ◦ φ − 1 ( w ) = φ  ( d 1 + d 2 ) φ − 1 ( w )  = φ  d 1 φ − 1 ( w )  + φ  d 2 φ − 1 ( w )  = φ ◦ α d 1 ◦ φ − 1 ( w ) + φ ◦ α d 2 ◦ φ − 1 ( w ) = e 1 w + e 2 w = ( e 1 + e 2 ) w . Then τ is injective , since if τ ( d ) = 0, then φ ◦ α d ◦ φ − 1 ( w ) = 0 for all w ∈ V ′ . This implies α d = 0, so d = 0. Rev erse the roles of D and D ′ and replace φ, σ by φ − 1 , σ − 1 . F or eac h k ∈ D ′ h , there is d ∈ D h suc h that φ − 1 ◦ α k ◦ φ = α d . F rom ab o ve there is τ ( d ) ∈ D ′ h suc h that φ ◦ α d ◦ φ − 1 = α τ ( d ) . Com bining these giv es α k = α τ ( d ) . Since k and τ ( d ) are homogeneous of the same degree in D ′ , it follows that k = τ ( d ), so τ is surjectiv e. Let d ∈ D h and v ∈ V . Then φ ( dv ) = φ ◦ α d ( v ) = α τ ( d ) ◦ φ ( v ) = τ ( d ) φ ( v ). Then using this we can show t ha t { u 1 , . . . , u k } are linearly indep enden t in V if and only if { φ ( u 1 ) , . . . , φ ( u k ) } are linearly indep enden t in V ′ . It follows that the f o rmer set spans V if and only if t he la tter set spans V ′ , pro ving dim D ( V ) = dim D ′ ( V ′ ).  Corollary 4.4.8. L et D , D ′ b e gr ade d division rings, a nd let ( d ) ∈ Γ n , ( d ′ ) ∈ Γ n ′ . If M n ( D )( d ) ∼ = gr M n ′ ( D ′ )( d ′ ) as gr ade d rings, then n = n ′ and D ∼ = gr D ′ . Pro of. As in the pro of of Prop osition 4.4.4 , w e can choose a graded D -mo dule V suc h that M n ( D )( d ) ∼ = gr End D ( V ), and a graded D ′ -mo dule V ′ suc h that M n ′ ( D ′ )( d ′ ) ∼ = gr Chapter 4. G r a ded Azuma y a Alg ebras 89 End D ′ ( V ′ ). Then b y Prop osition 4.4.7 , w e hav e that n = n ′ and D ∼ = gr D ′ .  Let D b e a Γ-gr a ded division ring and let ( λ ) = ( λ 1 , . . . , λ n ) ∈ Γ n . Consider the partit ion of Γ in to right cosets of Γ D . F o r distinct elemen ts of ( λ ), if the righ t cosets Γ D λ 1 , . . . , Γ D λ n are no t all distinct, then there is a ﬁrst Γ D λ i suc h that Γ D λ i = Γ D λ j for some j < i . In ( λ ), w e will replace λ i b y λ j , so that no w ( λ ) = ( λ 1 , . . . , λ i − 1 , λ j , λ i +1 , . . . , λ n ). If the righ t cosets Γ D λ 1 , . . . , Γ D λ i − 1 , Γ D λ i +1 , . . . , Γ D λ n are still not all distinct, rep eat the ab ov e pro cess. Con tin ue rep eating this pro cess un til all the right cosets of distinct elemen ts of ( λ ) are distinct. Let k denote t he n um b er of distinct righ t cosets (whic h is, aft er the ab o ve pro cess, also the num b er of distinct elemen ts in ( λ )). Let ε 1 = λ 1 , let ε 2 b e the second distinct elemen t of ( λ ), and so on until ε k is the k -th distinct elemen t of ( λ ). F or each ε l , let r l b e the n um b er of λ i in ( λ 1 , . . . , λ n ) with Γ D λ i = Γ D ε l . Using Prop osition 4.4.4 w e get M n ( D )( λ 1 , . . . , λ n ) ∼ = gr M n ( D )( ε 1 , . . . , ε 1 , ε 2 , . . . , ε 2 , . . . , ε k , . . . , ε k ) , (4.2) where eac h ε l o ccurs r l times. Also no te that f o r Γ- graded D -mo dules M and N , b y Prop o sition 4.2.6 , dim D ( M ⊕ N ) = dim D ( M ) + dim D  ( M ⊕ N ) / M  . By the ﬁrst isomorphism theorem, ( M ⊕ N ) / M ∼ = gr N , so dim D ( M ⊕ N ) = dim D ( M ) + dim D ( N ). The follo wing theorem extends [ 11 , Thm. 2.1] fro m trivially graded ﬁelds to graded division rings. Theorem 4.4.9. L et D b e a Γ -gr ade d division ring, let λ i , γ j ∈ Γ for 1 ≤ i ≤ n , 1 ≤ j ≤ m and let Ω = { λ 1 , . . . , λ n , γ 1 , . . . , γ m } . If the e lements o f Ω mutual ly c ommute and if M n ( D )( λ 1 , . . . , λ n ) ∼ = gr M m ( D )( γ 1 , . . . , γ m ) , (4.3) Chapter 4. G r a ded Azuma y a Alg ebras 90 then we ha v e n = m and γ i = τ i λ π ( i ) σ for e ach i , wher e τ i ∈ Γ D , π ∈ S n is a p ermutation and σ ∈ Z Γ (Ω) is ﬁxe d. Pro of. It follow s fro m Corollary 4.4.8 that n = m . As in ( 4.2 ), w e can ﬁnd ( ǫ ) = ( ε 1 , . . . , ε 1 , ε 2 , . . . , ε 2 , . . . , ε k , . . . , ε k ) in Γ n suc h tha t M n ( D )( λ 1 , . . . , λ n ) ∼ = gr M n ( D )( ǫ ). Let V = D ( ε − 1 1 ) ⊕ · · · ⊕ D ( ε − 1 1 ) ⊕ · · · ⊕ D ( ε − 1 k ) ⊕ · · · ⊕ D ( ε − 1 k ), a nd let e 1 , . . . , e n b e the standard homo g eneous basis of V . Then deg ( e i ) = ε s i , where ε s i is the i -th elemen t in ( ǫ ). Deﬁne E ij ∈ End D ( V ) b y E ij ( e l ) = δ il e j , for 1 ≤ i, j, l ≤ n . W e observ ed ab o v e (befor e Prop osition 4.4.4 ) that each E ij is a graded D - mo dule homomorphism of degree ε − 1 s i ε s j , the set { E ij : 1 ≤ i, j ≤ n } forms a ho mo g eneous basis for End D ( V ) a nd End D ( V ) ∼ = gr M n ( D )( ǫ ), w here E ij corresp onds to the matrix e ij in M n ( D )( ǫ ). F or an y i, j, h, l , w e ha v e E ij E hl = δ il E hj , so { E ii : 1 ≤ i ≤ n } forms a complete system of orthogonal idemp oten ts of End D ( V ). Similarly , w e can ﬁnd ( ǫ ′ ) = ( ε ′ 1 , . . . , ε ′ 1 , ε ′ 2 , . . . , ε ′ 2 , . . . , ε ′ k ′ , . . . , ε ′ k ′ ) ∈ Γ n suc h that M n ( D )( γ 1 , . . . , γ n ) ∼ = gr M n ( D )( ǫ ′ ). Let W = D ( ε ′ 1 − 1 ) ⊕ · · · ⊕ D ( ε ′ 1 − 1 ) ⊕ · · · ⊕ D ( ε ′ k ′ − 1 ) ⊕ · · · ⊕ D ( ε ′ k ′ − 1 ) and let e ′ 1 , . . . , e ′ n b e the standard homogeneous basis o f W with deg( e ′ i ) = ε ′ s i . W e ha v e End D ( W ) ∼ = gr M n ( D )( ǫ ′ ), so the gra ded isomorphism ( 4.3 ) pro vides a gra ded ring isomorphism θ : End D ( V ) → End D ( W ). Deﬁne E ′ ij := θ ( E ij ), for 1 ≤ i, j ≤ n , and let E ′ ii ( W ) = Q i for eac h i . Since { E ii : 1 ≤ i ≤ n } forms a complete system of orthogonal idempo ten ts, { E ′ ii : 1 ≤ i ≤ n } also forms a complete system of orthogonal idemp oten ts for End D ( W ). It fo llo ws that W = L 1 ≤ i ≤ n Q i . F or a n y i, j, h, l , as ab o ve , E ′ ij E ′ hl = δ il E ′ hj so E ′ ij E ′ j i = E ′ j j and E ′ ii acts as the iden tit y on Q i . By restricting E ′ ij to Q i , these relations induce a graded D - mo dule isomorphism E ′ ij : Q i → Q j of the same degree as E ij , na mely ε − 1 s i ε s j . So Q i ∼ = gr Q 1 ( ε − 1 s i ε 1 ) f o r any 1 ≤ i ≤ n , a nd it follo ws that W ∼ = gr L 1 ≤ i ≤ n Q 1 ( ε − 1 s i ε 1 ). Using the o bserv ations b efore the theorem regarding dimension count, it follo ws that dim D Q 1 = 1. So Q 1 is generated b y o ne homogeneous elemen t, sa y q , with Chapter 4. G r a ded Azuma y a Alg ebras 91 deg( q ) = α , and w e hav e Q 1 ∼ = gr D ( α − 1 ). Now Γ D α = Supp( D ( α − 1 )) = Supp( Q 1 ) ⊆ Supp( W ) = S i (Γ D ε ′ i ). But as the right cosets of Γ D in Γ are either disjoin t or equal, this implies Γ D α = Γ D ε ′ j for some j . Then Q 1 ∼ = gr D ( α − 1 ) = D ( ε ′ j − 1 τ − 1 j ) for some τ j ∈ Γ D . W e can easily sho w that D ( ε ′ j − 1 τ − 1 j ) ∼ = gr D ( ε ′ j − 1 ), so Q 1 ∼ = gr D ( ε ′ j − 1 ). Th us W ∼ = gr L 1 ≤ i ≤ n D ( ε − 1 s i ε 1 ε ′ j − 1 ). Then V = L 1 ≤ i ≤ n D ( ε − 1 s i ), so V ( ε 1 ε ′ j − 1 ) ∼ = gr M 1 ≤ i ≤ n D ( ε − 1 s i )( ε 1 ε ′ j − 1 ) = M 1 ≤ i ≤ n D ( ε 1 ε ′ j − 1 ε − 1 s i ) . Using the assumption that the elemen ts of Ω mutually commute, a nd since the elemen ts of b o t h ( ǫ ) and ( ǫ ′ ) are elemen ts of Ω, it fo llows that ε 1 ε ′ j − 1 ε − 1 s i = ε − 1 s i ε 1 ε ′ j − 1 . So we hav e W ∼ = gr V ( ε 1 ε ′ j − 1 ). Let σ = ε ′ j ε − 1 1 and denote this graded D -mo dule isomorphism b y φ : W → V ( σ − 1 ). Then φ ( e ′ i ) = P 1 ≤ j ≤ n a j e j , where a j ∈ D h and e j are homogeneous of degree ε s j σ in V ( σ − 1 ). Supp ose that deg( e j ) 6 = deg( e l ) for some j, l with a j , a l 6 = 0. Then since deg ( φ ( e ′ i )) = ε ′ s i = deg ( a j e j ) = deg( a l e l ) and ε s j σ 6 = ε s l σ , it follo ws that ε s j ε − 1 s l ∈ Γ D whic h con tra dicts the fact that Γ D ε s j and Γ D ε s l are distinct. So for all non-zero terms in the sum, each e j has the same degree. Th us ε ′ s i = τ j ε s j σ where τ j = deg( a j ) ∈ Γ D . Eac h ε s j w as chosen as ε s j = τ l λ l for some l with τ l ∈ Γ D , and similarly eac h ε ′ s i w as c hosen as ε ′ s i = τ h γ h for some h with τ h ∈ Γ D . Combining these giv es that γ i = τ i λ π ( i ) σ for 1 ≤ i ≤ n , where τ i ∈ Γ D and σ ∈ Z Γ (Ω).  Remark 4.4.10. W e note that fo r the con ve rse of the ab o ve theorem, w e need to assume that σ ∈ Z (Γ). Supp ose n = m and γ i = τ i λ π ( i ) σ for eac h i , where τ i ∈ Γ D , π ∈ S n is a p erm utation and σ ∈ Z (Γ) is ﬁxed. The n it follows immediately from Prop osition 4.4.4 that M n ( D )( λ 1 , . . . , λ n ) ∼ = gr M m ( D )( γ 1 , . . . , γ m ) . Giv en a g roup Γ and a ﬁeld K (whic h is not graded), it is a lso p ossible to deﬁne a grading on M n ( K ). Suc h a grading is deﬁned to b e a go o d gr ading of M n ( K ) if the matrices e ij are homogeneous, where e ij is the matr ix with 1 in the ij -p o sition. Chapter 4. G r a ded Azuma y a Alg ebras 92 These group gr a dings on matrix rings ha v e b een studied by D˘ asc˘ alescu et al [ 15 ]. The f o llo wing examples (from [ 15 , Eg. 1.3]) sho w tw o examples of Z 2 -grading on M 2 ( K ) f o r a ﬁeld K , one of whic h is a go o d grading; t he ot her is not a go o d grading. Examples 4.4.11. 1. L et R = M 2 ( K ) with the Z 2 grading deﬁned b y R [0] =      a 0 0 b   : a, b ∈ K    and R [1] =      0 c d 0   : c, d ∈ K    Then R is a graded ring with a go o d grading. 2. Let S = M 2 ( K ) with the Z 2 grading deﬁned by S [0] =      a b − a 0 b   : a, b ∈ K    and S [1] =      d c d − d   : c, d ∈ K    Then S is a graded ring, suc h that the Z 2 -grading is not a go o d grading , since e 11 is not homogeneous. Note that R and S are graded isomorphic, as the map f is an isomorphism: f : R − → S ;   a b c d   7− →   a + c b + d − a − c c d − c   So S do es not hav e a go o d grading, but it is isomorphic a s a graded ring to R whic h has a go o d grading. Let Γ be a gro up and let R b e a Γ-graded ring. If V is an n -dimens ional Γ-graded R -mo dule then, as ab o v e ( see § 4.2 and page 76 ), END R ( V ) = End R ( V ) fo r ms a Γ- graded ring. T aking R = K with the tr ivial Γ- grading, then V = L γ ∈ Γ V γ is just a v ector space with a Γ-g rading, for subspaces V γ of V . Then End K ( V ) ∼ = M n ( K ), whic h induces t he structure o f a Γ- graded K - algebra on M n ( K ). Supp ose { b 1 , . . . , b n } for ms a homogeneous basis for V , with deg ( b i ) = γ i . Then the matrix e ij ∈ M n ( K ) corresp onds to t he map E ij ∈ End K ( V ) deﬁned b y E ij ( b l ) = δ il b j , where δ is the Kronec k er delta. As E ij is homogeneous of degree Chapter 4. G r a ded Azuma y a Alg ebras 93 γ − 1 i γ j , e ij is also homogeneous, so this construction giv es a go o d Γ-grading on M n ( K ). This observ ation, combin ed with the follo wing prop osition, sho ws that a grading is go o d if and only if it can b e obtained in this manner from an n - dimensional Γ-graded K -v ector space. Prop osition 4.4.12. Consider a go o d Γ -gr ading on M n ( K ) . Then ther e exists a Γ -gr ade d ve ctor sp ac e V such that the is o morphism End( V ) ∼ = M n ( K ) , with r es p e c t to a h omo gen e ous b asis of V , is an isomorphism of Γ -gr ade d algebr as. Pro of. See [ 15 , Prop. 1.2].  4.5 Graded pro jectiv e mo dule s Let Γ b e a gr oup whic h is not necess arily abelian and R b e a Γ-graded ring. Through- out this section, unles s otherwise stated, we will assume that all g raded rings, graded mo dules and graded algebras are also Γ-g r a ded. W e will consider the catego ry R - gr - M od whic h is deﬁned as follows: the ob- jects are Γ-gra ded (left) R -mo dules, a nd for t wo ob jects M , N in R - gr - M od , the morphisms are deﬁned as Hom R - gr - M od ( M , N ) = { f ∈ Hom R ( M , N ) : f ( M γ ) ⊆ N γ for all γ ∈ Γ } . A graded R -mo dule P = L γ ∈ Γ P γ is said to b e gr a d e d pr oje c tive (resp. gr a de d fai th- ful ly pr oje ctive ) if P is pro jectiv e (resp. faithfully pro jectiv e) as an R -mo dule. W e use P gr( R ) to denote the subcatego ry of R - gr - M od consis ting of graded ﬁnitely generated pro jectiv e mo dules o ve r R . Then Prop osition 4.5.2 shows some equiv a- len t characterisations of gra ded pro jectiv e mo dules. Its pro of requires the follo wing prop osition, whic h is from [ 51 , Prop. 2.3.1]. Chapter 4. G r a ded Azuma y a Alg ebras 94 Prop osition 4.5.1. L et L, M , N b e gr ad e d R -mo dules, wi th R -line ar maps M g ! ! C C C C C C C C L h > > | | | | | | | | f / / N such that f = g ◦ h and f is a morphism in the c ate gory R - gr - M od . If g ( r esp. h ) is a morphism in R - gr - M od , then ther e exists a morphism h ′ : L → M (r esp. g ′ : M → N ) in R - gr - M od s uch that f = g ◦ h ′ (r esp. f = g ′ ◦ h ). Pro of. See [ 51 , Prop. 2.3.1].  Theorem 4.5.2. L et R b e a Γ -gr ade d ring and let P b e a gr ade d R -mo dule. The n the fol lowing a r e e q uiva lent: 1. P is gr ade d pr oje ctive; 2. F o r e a c h diag r am of gr ade d R -m o dule homomo rphisms P j   M g / / N / / 0 with g surje ctive, ther e is a g r ade d R -mo dule homomorphism h : P → M with g ◦ h = j ; 3. Hom R - gr - M od ( P , − ) is an exact functor in R - gr - M od ; 4. Every short exact se quenc e of gr a d e d R -mo dule hom omorphisms 0 − → L f − → M g − → P − → 0 splits via a gr a d e d map; 5. P is gr ade d isom o rphic to a dir e ct summand of a gr ade d fr e e R -mo dule F . Chapter 4. G r a ded Azuma y a Alg ebras 95 Pro of. (1) ⇒ (2 ) : Since P is pro jectiv e and g is a surjectiv e R -mo dule homomor- phism, t here is an R -mo dule homo mo r phism h : P → M with g ◦ h = j . By Prop osition 4.5.1 , there is a gra ded R -mo dule homomorphism h ′ : P → M with g ◦ h ′ = j . (2) ⇒ (3) : In exactly the same w ay as the non-graded setting (see [ 37 , Thm. IV.4.2]) w e can show tha t Hom R - gr - M od ( P , − ) is left exact. Then it follows immediately from (2) that it is righ t exact. (3) ⇒ (4): This is immediate. (4) ⇒ (5): Let { p i } i ∈ I b e a homogeneous generating set fo r P , where deg( p i ) = δ i . Let L i ∈ I R ( δ − 1 i ) b e the graded free R -mo dule with standard homog eneous basis { e i } i ∈ I where deg( e i ) = δ i . Then there is an exact seque nce 0 − → k er( g ) ⊆ − → M i ∈ I R ( δ − 1 i ) g − → P − → 0 , since the map g : L i ∈ I R ( δ − 1 i ) → P ; e i 7→ p i is a surjectiv e gr aded R -mo dule homo- morphism. By (4), there is a graded R - mo dule homomorphism h : P → L i ∈ I R ( δ − 1 i ) suc h that g ◦ h = id P . Since the exact sequence is, in part icular, a split exact sequence of R -mo dules, w e know from the non- graded setting [ 48 , Prop. 2.5] that t here is a n R - mo dule isomorphism θ : P ⊕ ker( g ) − → M i ∈ I R ( δ − 1 i ) ( p, q ) 7− → h ( p ) + q . Clearly t his map is also a graded R -mo dule homomorphism, so P ⊕ k er ( g ) ∼ = gr L i ∈ I R ( δ − 1 i ). (5) ⇒ (1): Graded free mo dules are free, so P is isomorphic to a direc t summand of a free R -mo dule. F r o m t he no n- graded setting, w e kno w that P is pro jectiv e.  Let Γ b e a n ab elian group, R b e a comm ut a tiv e Γ-gra ded ring, and w e deﬁne Chapter 4. G r a ded Azuma y a Alg ebras 96 m ultiplication in End R ( A ) to b e g · f = g ◦ f . Deﬁnition 4.5.3. A g r a ded R - a lgebra A is called a g r ade d Azumaya al g ebr a if the follo wing tw o conditions hold: 1. A is g raded faithfully pro jective ; 2. The natural map ψ A : A ⊗ R A op → End R ( A ) is a graded isomorphism. W e note that a gra ded R -a lgebra whic h is an Azumay a algebra (in the non-g raded sense) is a lso a graded Azuma y a algebra, since it is fa ithfully pro jectiv e as an R - mo dule, and the natural homomorphism A ⊗ R A op → End R ( A ) is clearly graded. So a graded central simple algebra o v er a graded ﬁeld (as in Theorem 4.3 .3 ) is in fact a graded Azuma y a algebra. The follo wing prop o sition pro v es, in the gr a ded se tting , a partial res ult of Morita equiv alence ( only in o ne direction), whic h w e will use in t he next c hapter ( see Propo - sition 5.4.3 ). F or the follo wing prop osition, w e require the group Γ to b e an ab elian group, so it will b e written additiv ely . W e o bserv e that if Γ is an ab elian group, with R a graded ring and ( d ) = ( δ 1 , . . . , δ n ) ∈ Γ n , then R n ( d ) is a graded M n ( R )( d )- R - bimo dule a nd R n ( − d ) is a g raded R - M n ( R )( d )- bimo dule. By [ 51 , p. 30], w e can deﬁne grading on the tensor pro duct of t w o g r aded mo dules in a similar w ay to that of graded algebras (as w e deﬁned on page 76 ). Prop osition 4.5.4 (Morita Equiv alence in the g raded setting) . L et Γ b e an a b elian gr oup, R b e a gr ade d ring and let ( d ) = ( δ 1 , . . . , δ n ) ∈ Γ n . Then the functors ψ : P gr( M n ( R )( d )) − → P gr( R ) P 7− → R n ( − d ) ⊗ M n ( R )( d ) P and ϕ : P gr( R ) − → P gr( M n ( R )( d )) Q 7− → R n ( d ) ⊗ R Q form e quivalenc es o f c ate gories. Chapter 4. G r a ded Azuma y a Alg ebras 97 Pro of. There are graded R - mo dule homomorphisms θ : R n ( − d ) ⊗ M n ( R )( d ) R n ( d ) − → R ( a 1 , . . . , a n ) ⊗ ( b 1 , . . . , b n ) 7− → a 1 b 1 + · · · + a n b n ; and σ : R − → R n ( − d ) ⊗ M n ( R )( d ) R n ( d ) a 7− → ( a, 0 , . . . , 0 ) ⊗ (1 , 0 , . . . 0) with σ ◦ θ = id and θ ◦ σ = id. F urther θ ′ : R n ( d ) ⊗ R R n ( − d ) − → M n ( R )( d )      a 1 . . . a n      ⊗      b 1 . . . b n      7− →      a 1 b 1 · · · a 1 b n . . . . . . a n b 1 · · · a n b n      and σ ′ : M n ( R )( d ) − → R n ( d ) ⊗ R R n ( − d ) ( m i,j ) 7− →         m 1 , 1 m 2 , 1 . . . m n, 1         ⊗         1 0 . . . 0         + · · · +         m 1 ,n . . . m n − 1 ,n m n,n         ⊗         0 . . . 0 1         are gra ded M n ( R )( d )- mo dule homomorphisms with σ ′ ◦ θ ′ = id and θ ′ ◦ σ ′ = id. So R n ( − d ) ⊗ M n ( R )( d ) R n ( d ) ∼ = gr R as graded R - R -bimo dules and R n ( d ) ⊗ R R n ( − d ) ∼ = gr M n ( R )( d ) as graded M n ( R )( d )- M n ( R )( d )- bimo dules resp ectiv ely . Then for P ∈ P gr( M n ( R )( d )), R n ( d ) ⊗ R R n ( − d ) ⊗ M n ( R )( d ) P ∼ = gr P . Supp ose f : P → P ′ is a graded M n ( R )( d )- mo dule homomorphism. Then there is a commutativ e diagram R n ( d ) ⊗ R R n ( − d ) ⊗ M n ( R )( d ) P id ⊗ f   / / P f   R n ( d ) ⊗ R R n ( − d ) ⊗ M n ( R )( d ) P ′ / / P ′ Chapter 4. G r a ded Azuma y a Alg ebras 98 F or Q ∈ P gr( R ), R n ( − d ) ⊗ M n ( R )( d ) R n ( d ) ⊗ R Q ∼ = gr Q . If g : Q → Q ′ is a graded R -mo dule homomorphism, then there is a comm utativ e diagram R n ( − d ) ⊗ M n ( R )( d ) R n ( d ) ⊗ R Q id ⊗ g   / / Q g   R n ( − d ) ⊗ M n ( R )( d ) R n ( d ) ⊗ R Q ′ / / Q ′ This shows t ha t there are natural equiv a lences from ψ ◦ ϕ to the identit y functor and from ϕ ◦ ψ to the iden tity . Hence ψ and ϕ are m utually in v erse equiv a lences of categories.  Chapter 5 Graded K -Theory of Azuma y a Algebras In Corollar y 4.3.4 , w e sa w that the K -theory of a graded cen tral simple algebra, graded by an ab elian group, is v ery close to the K - theory o f its centre, where this follo ws immediately from the corresp onding result in the no n- graded setting (see Theorem 3.1.5 ). Note that for the K -theory of a g raded central simple algebra A , w e are considering K i ( A ) = K i ( P r( A )), where P r( A ) denotes the category of ﬁnitely generated pro jective A -mo dules. But in the g raded setting, t here is also the category of graded ﬁnitely generated pro jectiv e mo dules ov er a giv en graded ring , whic h w e will consider in Section 5.4 . In Section 5.1 , w e consider the functor K 0 in the graded setting. F or a g r a ded ring R , w e set K gr 0 ( R ) = K 0 ( P gr( R )), where P gr( R ) is the catego r y of gra ded ﬁnitely generated pro jectiv e R -mo dules. W e sho w that this deﬁnition is equiv alen t to deﬁn- ing K gr 0 ( R ) to b e the group completion of P ro j gr ( R ), where P ro j gr ( R ) denotes the monoid of isomorphism classes of graded ﬁnitely generated pro jective mo dules o v er R . W e include a num b er of r esults in v olving K 0 of strongly g raded rings in Sec- tion 5.2 . Then in Section 5.3 , w e consider a sp eciﬁc example o f a graded Azuma y a algebra to sho w that t he graded K -t heory of this graded Azuma ya algebra is not the 99 Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 100 same as the graded K -theory of it s cen tre (see Example 5.3.2 ). So f or this example, in the setting of graded K -theory , Coro lla ry 4.3.4 do es not hold. W e also provide in Section 5.3 an example of a graded Azuma y a algebra suc h that its (no n-graded) K -theory do es not coincide with its graded K -theory . In Section 5.4 , we study the graded K -theory of graded Azuma y a algebras. W e in tro duce a n abstract functor called a graded D -f unctor deﬁned on the category of graded Azuma y a algebras g r a ded free ov er a ﬁxed comm utative graded r ing R (Deﬁnition 5.4.1 ). As in Section 3.1 , this allows us to sho w t ha t, for a graded Azuma y a a lgebra A graded free ov er R and sub ject to certain conditions, w e hav e a r elation similar to ( 3.1 ) in the gr a ded setting (see Theorem 5.4.4 and [ 3 2 ]). A corollary of this is that the isomorphism holds for an y Azuma y a alg ebra free o v er its cen tre (see Corollary 5.4.7 and Theorem 3.1.5 ). 5.1 Graded K 0 In Section 2.1 , w e deﬁned K 0 of a ring R to b e the g r o up completion o f the monoid P ro j( R ) o f isomorphism classes of ﬁnitely generated pro jectiv e R - mo dules. In [ 58 , Ch. 3], t here is another deﬁnition of K 0 , whic h deﬁnes K 0 for categories, rather than just for rings. Then using this deﬁnition of K 0 for categor ies, K 0 of a ring R can also b e deﬁned to b e K 0 ( P r( R )), whic h is sho wn in [ 58 , Thm. 3.1.7] to b e equiv alen t to the deﬁnition of K 0 giv en in Section 2.1 . In this section, w e let Γ b e a mu ltiplicative group and R b e a Γ-graded ring. W e b egin this section by stating the deﬁnition of a category with exact sequenc es. W e then show that P g r( R ) is a categor y with exact sequences , and w e r ecall from [ 58 , Deﬁn. 3.1.6] the deﬁnition of K 0 of a category (see Deﬁnition 5.1.2 ). Deﬁnition 5.1.1. A c ate gory with exact se quenc es is a full additiv e sub category C of an ab elian category A , with the following pro p erties: Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 101 1. C is closed under extensions; that is, if 0 − → P 1 − → P − → P 2 − → 0 is an exact sequenc e in A and P 1 , P 2 ∈ Ob j C , then P ∈ O b j C . 2. C has a small sk eleton; that is, C has a full sub category C 0 suc h that Ob j C 0 is a set, a nd for whic h the inclusion C 0 ֒ → C is an equiv alence. The exact sequences in suc h a category are deﬁned to b e the exact sequences in the am bien t category A in v olving only ob jects (and morphisms) a ll c hosen from C . Let R b e a g raded ring. W e will sho w that P gr ( R ) is a category with exact sequence s. Firstly no te that P gr( R ) is a full additiv e sub cat ego ry of R - gr - M od , where R - gr - M od is an ab elian catego r y ( by [ 51 , § 2.2]). T ak e 0 − → P 1 − → P − → P 2 − → 0 to b e an exact sequence in R - gr - M od . If P 1 , P 2 ∈ Ob j P gr( R ), t hen P 1 ⊕ Q 1 ∼ = gr R n ( δ 1 , . . . , δ n ) and P 2 ⊕ Q 2 ∼ = gr R m ( δ n +1 , . . . , δ n + m ), for some δ i ∈ Γ. So b y Theo- rem 4.5.2 , P ∼ = gr P 1 ⊕ P 2 from the exact sequence and P ⊕ ( Q 1 ⊕ Q 2 ) ∼ = gr ( P 1 ⊕ Q 1 ) ⊕ ( P 2 ⊕ Q 2 ) ∼ = gr R n + m ( δ 1 , . . . , δ n + m ); that is, P ∈ Ob j P gr( R ). F or the second prop erty , ta k e C 0 to b e the set of graded direct summands of { R n ( d ) : n ∈ N , ( d ) ∈ Γ n } . W e will observ e here that this is a set. G raded direct summands are, in particular, direct summands, a nd w e kno w from the no n-graded setting that the collection of direct summands of { R n : n ∈ N } is a set. T aking the union of these sets o ve r all ( d ) ∈ Γ n , we still hav e a set since the union of sets indexed b y a set is still a set. F or the s econd part of prop ert y (2), w e hav e an equiv alence o f categories from C 0 to P g r ( R ). This follo ws sinc e for an y P ∈ P gr( R ), w e kno w from T heorem 4.5.2 that Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 102 P is graded isomorphic to a graded direct summand of { R n ( d ) : n ∈ N , ( d ) ∈ Γ n } . This giv es a f unctor F : P gr( R ) → C 0 . Consider the inclusion functor J : C 0 → P gr( R ). Then it is routine to c hec k that the functors F and J form m utually inv erse equiv alences of categories. Deﬁnition 5.1.2. Let C b e a category with exact sequences with small sk eleton C 0 . Then K 0 ( C ) is deﬁned to b e the free a b elian gro up based on Ob j C 0 , mo dulo t he follo wing relations: 1. [ P ] = [ P ′ ] if there is an isomorphism P ∼ = P ′ in C . 2. [ P ] = [ P 1 ] + [ P 2 ] if there is a short exact sequence 0 → P 1 → P → P 2 → 0 in C . Here [ P ] denotes the elemen t o f K 0 ( C ) corresp onding to P ∈ Ob j C 0 . W e note that relation (1) is a special case of (2) with P 1 = 0. Since ev ery P ∈ Ob j C is isomorphic to an ob ject of C 0 , the notation [ P ] makes sense for ev ery ob ject of C . W e will now o bserv e that P ro j gr ( R ), the isomorphism classes of gr a ded ﬁnitely generated pro jectiv e mo dules, forms a monoid. Firstly it is a set, b y the ab ov e argumen t, since it is the set of direct summands of { R n ( d ) : n ∈ N , ( d ) ∈ Γ n } , mo dulo the equiv alence relation of g raded isomorphism. W e will show the direct sum o n P r o j gr ( R ) is w ell deﬁned. If [ P ] = [ P ′ ] and [ Q ] = [ Q ′ ], then P ∼ = gr P ′ and Q ∼ = gr Q ′ . So w e ha v e P ⊕ Q ∼ = gr P ′ ⊕ Q ′ ; that is, [ P ] + [ Q ] = [ P ′ ] + [ Q ′ ]. Clearly the binary op eratio n is commutativ e and asso ciativ e, and the identit y elemen t in P ro j gr ( R ) is the isomorphism class of the zero mo dule. F or a graded ring R , w e deﬁne K gr 0 ( R ) = K 0 ( P gr( R )) , where K 0 ( P gr( R )) is deﬁned to be K 0 of the category P gr( R ) . W e sho w in Theo- rem 5.1.3 that this de ﬁnition o f K gr 0 is equiv alent to deﬁn ing K gr 0 ( R ) to b e the group completion of P ro j gr ( R ) (see [ 58 , Thm. 1.1 .3] for the group completion construction). Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 103 The following pro of follows in a similar wa y to the equiv alen t result in the non-g raded setting (see [ 58 , Thm. 3.1.7]). Theorem 5.1.3. L et R b e a gr ade d ring and let P gr( R ) b e the c ate gory o f gr ade d ﬁnitely gener ate d pr oje ctive mo dules over R . T hen the gr oup c ompletion of P ro j gr ( R ) may b e identiﬁe d natur al ly with K 0 ( P gr( R )) , wher e K 0 ( P gr( R )) i s deﬁne d in Deﬁ- nition 5.1.2 . Pro of. By deﬁnition, K 0 ( P gr( R )) and the group completion of P ro j gr ( R ) are b oth deﬁned to b e ab elian groups with one g enerator for e ach isomorphism class of g raded ﬁnitely generated pro jectiv e mo dules ov er R . Addition in the latter is de ﬁned a s [ P ] + [ Q ] = [ P ⊕ Q ]. In K 0 ( P gr( R )), [ P ] + [ Q ] is deﬁned t o b e [ N ] for a graded ﬁnitely generated pro jectiv e R -mo dule N if there is an exact sequenc e 0 − → P − → N − → Q − → 0 in P gr( R ). If N = P ⊕ Q then there is clearly an exact sequence 0 − → P − → P ⊕ Q − → Q − → 0 in P gr( R ). Th us [ P ⊕ Q ] = [ N ], and t he a ddition op erations in the tw o groups coincide. W e also need to c hec k that b oth gro ups satisfy the same relations. The g r oup completion construction is the free ab elian gr o up ba sed on P ro j gr ( R ) sub ject to the relations [ P ] = [ P ′ ] if P ∼ = gr P ′ and [ P ] + [ Q ] = [ P ⊕ Q ]. W e hav e observ ed ab ov e that the second relatio n holds in K 0 ( P gr( R )), and it is clear that the ﬁrst relation also holds. Then K 0 ( P gr( R )) is sub ject to the relations (1) and (2) as in Deﬁnition 5.1.2 . It remains to c heck that the second of these relations holds in the gr o up completion construction. Supp ose 0 − → P 1 f − → P g − → P 2 − → 0 Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 104 is an exact sequence in P gr( R ). By Theorem 4.5.2 this is split exact since P 2 is graded pro jectiv e. So there is a gra ded R -mo dule homomorphism h : P 2 → P with g ◦ h = id P 2 . Then as in the pro of of Theorem 4.5.2 , there is a g r aded R - mo dule isomorphism P 1 ⊕ P 2 − → P ( p, q ) 7− → f ( p ) + h ( q ) . So with the group completion construction, [ P ] = [ P 1 ⊕ P 2 ] = [ P 1 ] + [ P 2 ] whic h sho ws that it satisﬁes the second relation.  W e also observ e that b y [ 58 , p. 291], for i = 0, Quillen’s Q -construction of K 0 for a category (as in Section 2.3 ) coincides with D eﬁnition 5.1.2 of K 0 of a cat ego ry . W e ﬁnish this section b y calculating K gr 0 of a trivially graded ﬁeld. Prop osition 5.1.4. L et F b e a ﬁeld, Γ b e a gr oup and c onsider F as a trivial ly Γ -gr ade d ﬁeld. Then K gr 0 ( F ) ∼ = L γ ∈ Γ Z γ wher e Z γ = Z fo r e ac h γ ∈ Γ . Pro of. Let M b e a graded ﬁnitely generated pro jectiv e F -mo dule. By Theorem 4.2.4 , M is graded free so M ∼ = gr F ( δ 1 ) r 1 ⊕ · · · ⊕ F ( δ k ) r k , where r i ∈ N and the δ i ∈ Γ are distinct. T o show tha t M is written uniquely in this w ay , supp ose M ∼ = gr F ( α 1 ) r ′ 1 ⊕ · · · ⊕ F ( α l ) r ′ l for some r ′ i ∈ N and some distinct α i ∈ Γ. Consider the set { γ 1 , . . . , γ n } = { δ 1 , . . . , δ k } ∪ { α 1 , . . . , α l } . By rearranging the terms and adding zeros where required, w e hav e F ( δ 1 ) r 1 ⊕ · · · ⊕ F ( δ k ) r k = F ( γ 1 ) s 1 ⊕ · · · ⊕ F ( γ n ) s n where s i ∈ N and the γ i are distinct. Similarly , F ( α 1 ) r ′ 1 ⊕ · · · ⊕ F ( α l ) r ′ l can b e written as F ( γ 1 ) s ′ 1 ⊕ · · · ⊕ F ( γ n ) s ′ n . W e note that as F e -mo dules  F ( γ 1 ) s 1 ⊕ · · · ⊕ F ( γ n ) s n  γ − 1 1 ∼ =  F ( γ 1 ) s ′ 1 ⊕ · · · ⊕ F ( γ n ) s ′ n  γ − 1 1 . Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 105 If s 1 = 0, then on the left hand side of the isomorphism we ha v e zero, since F ( γ i ) γ − 1 1 = 0 for all i 6 = 1 . So w e m ust also ha v e zero on the right hand s ide of the isomorphism. Therefore as F ( γ 1 ) γ − 1 1 = F e = F , w e m ust hav e s ′ 1 = 0. If s 1 6 = 0, then on the left hand side of the isomorphism, w e ha v e F s 1 . Since it is a n F -mo dule isomorphism, w e ha v e the same on the righ t hand side, and thus s 1 = s ′ 1 . Rep eat the same argumen t fo r eac h γ i ∈ { γ 1 , . . . , γ n } . This shows that for eac h i , w e ha v e s i = s ′ i . Th us M can b e written uniqu ely as F ( δ 1 ) r 1 ⊕ · · · ⊕ F ( δ k ) r k , where r i ∈ N and t he δ i ∈ Γ are distinct. This giv es an isomor phism from P ro j gr ( F ) to L γ ∈ Γ N γ where N γ = N for each γ ∈ Γ. As the group completion of N is Z , it f ollo ws that K gr 0 ( F ) is isomorphic to L γ ∈ Γ Z γ where Z γ = Z for eac h γ ∈ Γ.  5.2 Graded K 0 of strongly grade d ri n gs Throughout this section, w e let Γ b e a multiplic ative group and R b e a Γ-g raded ring. F or an y R e -mo dule N and a ny γ ∈ Γ, w e identify the R e -mo dule R γ ⊗ R e N with its image in R ⊗ R e N . Then R ⊗ R e N is a Γ-gr a ded R -mo dule, with R ⊗ R e N = L γ ∈ Γ R γ ⊗ R e N . Consider the restriction functor G : R - gr - M od − → R e - M od M 7− → M e ψ 7− → ψ | M e , and the induction functor deﬁned by I : R e - M od − → R - gr - M od N 7− → R ⊗ R e N φ 7− → id R ⊗ φ. Prop osition 5.2.1 and Theorem 5.2.2 are from Da de [ 14 , p. 245] (see also [ 5 1 , Thm. 3.1.1]). Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 106 Prop osition 5.2.1. L et R b e a Γ -gr ade d ring. With G a nd I d e ﬁne d as ab o v e, ther e is a natur al e quiva lenc e of the c omp osite functor G ◦ I : R e - M od → R e - M od with the identity f unctor on R e - M od . Pro of. Let N ∈ R e - M od . Then G ◦ I ( N ) = ( R ⊗ R e N ) e = R e ⊗ R e N . W e kno w that for the ring R e , the map α : R e ⊗ R e N → N ; r ⊗ n 7→ r n is a n isomorphism. F o r φ : N → N ′ in R e - M od , the follo wing diagra m is clearly comm utativ e R e ⊗ R e N α / / id R e ⊗ φ   N φ   R e ⊗ R e N ′ α / / N ′ So α is a natural equiv alence from G ◦ I to the iden tity f unctor.  Theorem 5.2.2 (D ade’s Theorem) . L et R b e a Γ -gr ade d ring. If R is str ongly gr ade d, then the functors G and I deﬁne d ab o ve form mutual ly inverse e quiva l e nc es of c ate gories . Pro of. Let M b e a graded R - mo dule. Then I ◦ G ( M ) = R ⊗ R e M e . W e will sho w that the natural map β : R ⊗ R e M e → M ; r ⊗ m 7→ r m is a g raded R -mo dule isomorphism. It is an R - mo dule homomorphism us ing prop erties of tensor pro ducts, and is clearly graded. Since R is strongly g r a ded, it follo ws that for all γ , δ ∈ Γ, M γ δ = R e M γ δ = R γ R γ − 1 M γ δ ⊆ R γ M δ ⊆ M γ δ so w e hav e R γ M δ = M γ δ . W e note that β ( R γ ⊗ R e M e ) = R γ M e = M γ , so β is surjectiv e. Let N = k er ( β ), whic h is a graded R -submo dule of R ⊗ R e M e , so N e = N ∩ ( R e ⊗ R e M e ). Now N e = k er( α ), where α : R e ⊗ R e M e → M e is the canonical isomorphism, so N e = 0. Since N is a graded R -mo dule, as ab o v e w e ha v e N γ = R γ N e = 0 fo r all γ ∈ Γ. It follows that β is injectiv e. Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 107 Let ψ : M → M ′ in R - gr - M od . Then the following diagra m commutes R ⊗ R e M e β / / id R ⊗ ψ | M e   M ψ   R ⊗ R e M ′ e β / / M ′ so β is a natural equiv alence from I ◦ G to the iden tity func tor , completing the pro o f.  Prop osition 5.2.3. L et R b e a Γ -gr ade d ring. If R is str ongly gr ade d, then for e ach γ ∈ Γ , R γ is a ﬁnitely gene r ate d pr oje ctive left (or right) R e -mo dule. Pro of. See [ 50 , Cor. 2.16.10].  W e no w show t ha t the f unctor s G a nd I , when restricted to the categories of ﬁnitely generated pro jectiv e mo dules, still f o rm an equiv alence of categories. Corollary 5.2.4. L et R b e a Γ -gr ade d ring. If R is str ongly gr ade d , t hen the functors G : P gr( R ) − → P r( R e ) and I : P r( R e ) − → P gr( R ) form m utual ly inverse e quivalenc es of c ate gories. Pro of. If A ∈ P r( R e ), then A ⊕ B ∼ = R n e for some R e -mo dule B . Then I ( A ⊕ B ) ∼ = I ( R n e ), so I ( A ) ⊕ I ( B ) ∼ = R n . This sho ws I ( A ) is ﬁnitely generated and pro jectiv e as an R -mo dule, and we kno w I ( A ) ∈ R - gr - M od , so I ( A ) ∈ P gr( R ). If M ∈ P g r( R ), then M ⊕ N ∼ = gr R n ( d ) for some graded R -mo dule N and some ( d ) = ( δ 1 , . . . , δ n ) ∈ Γ n . Then G ( M ⊕ N ) ∼ = G ( R n ( d )), so G ( M ) ⊕ G ( N ) ∼ = R δ 1 ⊕ · · · ⊕ R δ n . Since, b y Prop osition 5.2.3 , each R δ i is a ﬁnitely generated pro jectiv e mo dule o v er R e , we hav e that G ( M ) is also ﬁnitely generated and pro jective as an R e -mo dule. The result now follows from Theorem 5.2.2 .  W e note that I and G as in Corollary 5.2.4 are exact functors b et w een exact categories. This fo llo ws since if 0 → L → M → N → 0 is an exact sequenc e in Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 108 P gr( R ), then as N is graded pro jectiv e, the exact sequence splits. So M ∼ = gr L ⊕ N and G ( M ) ∼ = G ( L ⊕ N ) ∼ = G ( L ) ⊕ G ( N ). Th us 0 → G ( L ) → G ( M ) → G ( N ) → 0 is a split exact sequence in P r( R e ). Supp ose 0 → A → B → C → 0 is a n exact sequence in P r( R e ). Then the exact sequence splits, so B ∼ = A ⊕ C a nd I ( B ) ∼ = gr I ( A ⊕ C ) ∼ = gr I ( A ) ⊕ I ( C ). Deﬁne R -mo dule homomorphisms π : I ( A ) ⊕ I ( C ) − → I ( C ) and ı : I ( A ) − → I ( A ) ⊕ I ( C ) ( a, c ) 7− → c a 7− → ( a, 0) . F rom the non-graded setting [ 48 , Prop. 2.7], 0 / / I ( A ) θ − 1 ◦ ı / / I ( B ) π ◦ θ / / I ( C ) / / 0 is an exact sequence of R -mo dules, where θ : I ( B ) → I ( A ) ⊕ I ( C ) is the R -mo dule homomorphism a s ab ov e. Then as the maps θ , ı, π are g r a ded R - mo dule homomor- phisms, the exact sequence is an exact sequence in P gr( R ). Prop osition 5.2.5. L et R b e a Γ -gr ade d ring. If R is str on g ly gr ade d, then K gr 0 ( R ) ∼ = K 0 ( R e ) . Pro of. Since R is strongly graded, w e can a pply Corollary 5.2.4 , which sa ys that t he category of graded ﬁnitely generated pro j ective mo dules ov er R is equiv alent to the category of ﬁnitely generated pro jectiv e mo dules o v er R e . By Theorem 2.3.2 , eac h K i is a functor fro m the catego ry o f exact catego ries with exact functors to the category of abelian groups. W e observ ed ab ov e that I and G are exact functors betw een exact categories, so this implies K 0  P gr( R )  ∼ = K 0  P r( R e )  as a b elian groups. That is, using the previous notation, K gr 0 ( R ) ∼ = K 0 ( R e ).  Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 109 5.3 Examples In this section, w e consider a sp eciﬁc example of a graded Azuma y a alg ebra A (see Example 5.3.2 ). W e show tha t K gr 0 ( A ) ⊗ Z [1 /n ] is not isomorphic to K gr 0 ( Z ( A )) ⊗ Z [1 /n ] for this graded Azuma y a algebra A . This leads to the follow ing question, whic h w e will partially answ er in Section 5.4 (see Theorem 5.4.4 ). Question 5.3.1. Let Γ b e an ab elian group, R b e a commutativ e Γ-graded ring, and A b e a graded Azuma y a algebra ov er its cen tre R of rank n . When do w e ha ve K gr 0 ( A ) ⊗ Z [1 /n ] ∼ = K gr 0 ( R ) ⊗ Z [1 /n ]? W e no w explain the example mentioned ab o v e. Example 5.3.2. Consider the quaternion algebra H = R ⊕ R i ⊕ R j ⊕ R k . In Example 1.5.4 , we sho w ed that H is an Azuma ya a lgebra o ve r R . Then fro m Exam- ple 4.2.3 (2), H is a Z 2 × Z 2 -graded division ring, so it is in fact a graded Azuma y a algebra, whic h is strongly Z 2 × Z 2 -graded. W e can no w use Prop osition 5.2.5 . Here H 0 = R , so K gr 0 ( H ) ∼ = K 0 ( R ) ∼ = Z . The cen tre Z ( H ) = R is a ﬁeld and is trivially g raded b y Z 2 × Z 2 . By Prop osition 5.1.4 , K gr 0  Z ( H )  = K gr 0 ( R ) ∼ = Z ⊕ Z ⊕ Z ⊕ Z . W e can see that K gr 0 ( H ) ⊗ Z [1 / 2] ∼ = Z ⊗ Z [1 / 2], but K gr 0  Z ( H )  ⊗ Z [1 / 2] ∼ = ( Z ⊕ Z ⊕ Z ⊕ Z ) ⊗ Z [1 / 2], so they ar e clearly not isomorphic. W e g ive here another example, whic h generalises the ab ov e example of H a s a Z 2 × Z 2 -graded ring. Example 5.3.3. Let F b e a ﬁeld, ξ b e a primitiv e n -th ro ot of unit y and let a , b ∈ F ∗ . Let A = n − 1 M i =0 n − 1 M j =0 F x i y j b e the F -algebra generated b y the elemen ts x and y , whic h a re sub ject to t he relat io ns x n = a , y n = b and xy = ξ y x . By [ 20 , Thm. 11.1], A is an n 2 -dimensional cen t ral simple algebra o v er F . Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 110 F urther, we will show that A forms a graded division ring. Clearly A can b e written as a direct sum A = M ( i,j ) ∈ Z n ⊕ Z n A ( i,j ) , where A ( i,j ) = F x i y j and eac h A ( i,j ) is an additiv e subgroup of A . Using the fact tha t ξ − k j x k y j = y j x k for eac h j, k , with 0 ≤ j, k ≤ n − 1, w e can show that A ( i,j ) A ( k, l ) ⊆ A ([ i + k ] , [ j + l ]) , fo r i, j, k , l ∈ Z n . A non-zero homogeneous elemen t f x i y j ∈ A ( i,j ) has a n in v erse f − 1 a − 1 b − 1 ξ − ij x n − i y n − j , pro ving A is a graded division ring. Clearly the suppo rt of A is Z n × Z n , so A is strongly Z n × Z n -graded. As for Example 5.3.2 , K gr 0 ( A ) ∼ = K 0 ( A 0 ) = K 0 ( F ) ∼ = Z . The ce ntre F is trivially graded b y Z n × Z n , so K gr 0  Z ( A )  ∼ = L n 2 i =1 Z i where Z i = Z . Remark 5.3.4. W e sa w in Example 4.2.3 (2) that H can a lso b e considered as a Z 2 -graded division ring. So H is also strongly Z 2 -graded, and K gr 0 ( H ) ∼ = K 0 ( H 0 ) = K 0 ( C ) ∼ = Z . Then Z ( H ) = R , whic h w e can consider as a trivially Z 2 -graded ﬁeld, so b y Prop osition 5.1.4 , K gr 0 ( R ) = Z ⊕ Z . W e note that fo r b oth grade groups, Z 2 and Z 2 × Z 2 , w e ha v e K gr 0 ( H ) ∼ = Z , but the K gr 0 ( R ) are diﬀerent. So the graded K -theory of a graded ring dep ends not only on the ring, but also on its grade gro up. W e note that in the ab ov e example, the graded K - theory of H is isomorphic to the usual K -theory of H . This follows since H is a division ring, so b y Example 2.1.5 (1) w e hav e K 0 ( H ) ∼ = Z , and w e observ ed ab ov e t hat K gr 0 ( H ) ∼ = Z . But it is no t alw ay s the case that the graded K -theory and usual K -t heory coincide. F or R , w e kno w that K 0 ( R ) ∼ = Z . But in the ab ov e examples, when R w as trivially graded b y Z 2 (resp. Z 2 × Z 2 ), w e had K gr 0 ( R ) ∼ = Z ⊕ Z (resp. K gr 0 ( R ) ∼ = Z ⊕ Z ⊕ Z ⊕ Z ). Below w e will giv e an example o f a graded ring which is not trivially graded, and its graded K -theory is not isomorphic to its usual K -theory . Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 111 Example 5.3.5. Let K b e a ﬁeld and let R = K [ x 2 , x − 2 ]. Then R is a Z -graded ﬁeld, where R can b e written as R = L n ∈ Z R n , with R n = K x n if n is ev en and R n = 0 if n is o dd. Conside r the shifted graded matrix ring A = M 3 ( R )(0 , 1 , 1), whic h has supp ort Z . Then we will sho w that A is a graded cen tral simple a lg ebra o v er R , so b y Theorem 4.3.3 , A is a graded Azuma y a algebra ov er R . It is clear that the cen tre of A is R , and A is ﬁnite dimensional ov er R . W e note that if A has a non-zero homogeneous t w o-sided ideal J , the n J is generated b y homogeneous elemen ts (see Prop osition 4.1.4 ). Using the elemen tary matrices, w e can sho w that J = A (see [ 37 , Ex. I I I.2.9]), so A is graded simple. F urther, we will sho w that A is a strongly Z -graded ring. Using Prop osition 4.1.3 , it is suﬃcien t to sho w that I 3 ∈ A n A − n for all n ∈ Z . T o sho w this, w e note that I 3 =      0 1 0 0 0 0 0 0 0           0 0 0 1 0 0 0 0 0      +      0 0 0 x 2 0 0 0 0 0           0 x − 2 0 0 0 0 0 0 0      +      0 0 0 0 0 0 x 2 0 0           0 0 x − 2 0 0 0 0 0 0      and I 3 =      0 x − 2 0 0 0 0 0 0 0           0 0 0 x 2 0 0 0 0 0      +      0 0 0 1 0 0 0 0 0           0 1 0 0 0 0 0 0 0      +      0 0 0 0 0 0 1 0 0           0 0 1 0 0 0 0 0 0      So I 3 ∈ A 1 A − 1 and I 3 ∈ A − 1 A 1 , and the required result follows b y induction, sho wing A is strongly graded. As in the previous examples, using Prop osition 5.2.5 we ha v e K gr 0 ( A ) ∼ = K 0 ( A 0 ). Here R 0 = K , so there is a ring isomorphism A 0 =      R 0 0 0 0 R 0 R 0 0 R 0 R 0      ∼ = K × M 2 ( K ) . Then K gr 0 ( A ) ∼ = K 0 ( A 0 ) ∼ = K 0 ( K ) ⊕ K 0  M 2 ( K )  ∼ = Z ⊕ Z , Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 112 since K 0 resp ects Cartesian pro ducts and Morita equiv alence. Note that for the usual K -theory of A , K 0  M 3 ( R )(0 , 1 , 1)  ∼ = K 0 ( R ) = K 0  K [ x 2 , x − 2 ]  . Using the F undamen tal Theorem of Algebraic K -theory [ 58 , Thm. 3.3.3] (see also [ 48 , p. 484]), K 0  K [ x, x − 1 ]  ∼ = K 0 ( K ) ∼ = Z , and it follow s that K 0 ( A ) ∼ = Z , since K [ x 2 , x − 2 ] ∼ = K [ x, x − 1 ] as rings. So the K -theory of A is isomorphic to one cop y of Z , which is not the same as the graded K -theory of A . 5.4 Graded D -functors Throughout this section, w e will assume that Γ is an ab elian group, R is a ﬁxed comm utativ e Γ-graded ring and all graded rings, graded mo dules and graded algebras are also Γ-graded. As men tioned in Remark 4.4.3 , in this section we will deﬁne m ultiplication in End R ( A ) to b e g · f = g ◦ f . Let A b b e the category of ab elian groups and let Az gr ( R ) denote the category of graded Azuma y a a lg ebras graded free o v er R with graded R -alg ebra homomorphisms. W e recall from Section 4.2 that Γ ∗ M k ( R ) =  ( d ) ∈ Γ k : GL k ( R )[ d ] 6 = ∅  , where, for ( d ) = ( δ 1 , . . . , δ k ) ∈ Γ k , GL k ( R )[ d ] consists of in v ertible k × k matr ices with the ij -en try in R − δ i (see page 71 ). Deﬁnition 5.4.1. An abstract functor F : Az gr ( R ) → A b is deﬁned to b e a gr ade d D -functor if it satisﬁes the three prop erties b elow: (1) F ( R ) is the trivial gro up. (2) F or an y graded Azuma ya algebra A graded free ov er R and for an y ( d ) = ( δ 1 , . . . , δ k ) ∈ Γ ∗ M k ( R ) , there is a homomorphism ρ : F  M k ( A )( d )  − → F ( A ) suc h that the composition F ( A ) → F  M k ( A )( d )  → F ( A ) is η k , where η k ( x ) = Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 113 x k . (3) With ρ as in pr o p ert y (2), then k er( ρ ) is k - torsion. Note that these prop erties are w ell-deﬁned since bo th R and M k ( A )( d ) are gra ded Azuma y a algebras graded f ree ov er R . The pro of of the theorem b elow follo ws in a similar w a y to that of Theorem 3.1.2 . Theorem 5.4.2. L et A b e a gr ade d Azumaya algebr a which is gr ade d fr e e over its c en tr e R of dimension n , such that A has a h omo gen e ous b asis with de gr e es ( δ 1 , . . . , δ n ) in Γ ∗ M n ( R ) . Then F ( A ) is n 2 -torsion, w her e F is a gr ade d D -func tor. Pro of. Let { a 1 , . . . , a n } b e a homog eneous basis for A ov er R , and let ( d ) = (deg ( a 1 ) , . . . , deg( a n )) ∈ Γ ∗ M n ( R ) . Since R is a graded Azuma ya algebra o ver itself, by (2 ) in the de ﬁnition of a graded D -functor, there is a homomorphism ρ : F  M n ( R )( d )  → F ( R ). But F ( R ) is trivial b y prop erty (1) and therefore the k ernel of ρ is F  M n ( R )( d )  whic h is, b y (3), n -torsion. F urther, the graded R -algebra isomorphism A ⊗ R A op ∼ = gr End R ( A ) from the deﬁnition of a graded Azuma y a algebra, combined with the graded isomorphism End R ( A ) ∼ = gr M n ( R )( d ), induces a n isomorphism F ( A ⊗ R A op ) ∼ = F ( M n ( R )( d )). So F ( A ⊗ R A op ) is also n -torsion. In the category Az gr ( R ), the tw o graded R -algebra homomorphisms i : A → A ⊗ R A op and r : A op → End R ( A op ) → M n ( R )( d ) induce group homomorphisms i : F ( A ) → F ( A ⊗ R A op ) and r : F ( A ⊗ R A op ) → F ( A ⊗ R M n ( R )( d )), where F ( A ⊗ R M n ( R )( d )) ∼ = F ( M n ( A )( d )). Consider the follow ing diagram F ( A ) i   η n   F ( A ⊗ R A op ) r   F ( M n ( A )( d )) ρ / / F ( A ) whic h is comm utative b y prop erty (2). It f o llo ws that F ( A ) is n 2 -torsion.  Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 114 F or a gra ded ring A , let P g r( A ) b e the categor y of graded ﬁnitely generated pro jectiv e A - mo dules. Then as for K gr 0 in Section 5.1 , w e deﬁne K gr i ( A ) = K i ( P gr( A )) for i ≥ 0, where K i ( P gr( A )) is the Quillen K -group of the exact categor y P gr( A ) (see Section 2.3 ). Let A b e a graded ring with graded cen tre R . Then the g raded ring homo mo r - phism R → A induces an exact f unctor A ⊗ R − : P gr( R ) → P gr( A ), whic h in turn induces a group homomorphism K gr i ( R ) → K gr i ( A ). Then w e ha ve an exact sequence 1 − → ZK gr i ( A ) − → K gr i ( R ) − → K gr i ( A ) − → CK gr i ( A ) − → 1 (5.1) where ZK gr i ( A ) and CK gr i ( A ) are the k ernel a nd cok ernel of the map K gr i ( R ) → K gr i ( A ) resp ectiv ely . W e will sho w that CK gr i can b e regarded a s the follo wing functor CK gr i : Az gr ( R ) − → A b A 7− → CK gr i ( A ) . F or a g raded Azuma ya algebra A graded free o v er R , clearly CK gr i ( A ) = cok er  K gr i ( R ) − → K gr i ( A )  is an ab elian group. Consider graded Azuma y a algebras A, A ′ graded free o v er R and a graded R -alg ebra homomorphism f : A → A ′ . Then there is an induced exact functor A ′ ⊗ A − : P gr( A ) → P gr( A ′ ), which induces a group homomorphism f ∗ : K gr i ( A ) → K gr i ( A ′ ). W e ha ve a n exact functor A ′ ⊗ A ( A ⊗ R − ) : P gr( R ) − → P gr( A ) − → P gr( A ′ ) . As the map f restricted to R is the iden tity map, the induced functor P gr( R ) → P gr( R ) is also t he iden tity . So it induces the iden tit y map K gr i ( R ) → K gr i ( R ). W e Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 115 ha v e an exact functor A ′ ⊗ R − : P gr( R ) − → P gr( R ) − → P gr( A ′ ) . Since these t wo functors from P gr( R ) to P g r( A ′ ) are isomorphic, they induce the same map on the lev el of the K -g roups by Theorem 2.3.2 . Since K i is a functor from the category of ex act catego r ies to the abelian g r oups, the follo wing diagr a m is commutativ e K gr i ( R ) / / id   K gr i ( A ) / / f ∗   CK gr i ( A ) CK gr i ( f )   K gr i ( R ) / / K gr i ( A ′ ) / / CK gr i ( A ′ ) Then it is routine to c hec k that CK gr i forms the r equired functor. Similar ly ZK gr i can b e regarded as the follo wing functor ZK gr i : Az gr ( R ) − → A b A 7− → ZK gr i ( A ) . Prop osition 5.4.3. With CK gr i deﬁne d as ab ove, CK gr i is a gr ade d D -functor. Pro of. Prop erty (1) is clear, since R is comm utative so K gr i ( Z ( R )) → K gr i ( R ) is the iden tit y map. F o r prop ert y (2), let A b e a graded Azuma ya algebra graded free o v er R and let ( d ) = ( δ 1 , . . . , δ k ) ∈ Γ ∗ M k ( R ) . Then there are functors: φ : P gr( A ) − → P gr( M k ( A )( d )) (5.2) X 7− → M k ( A )( d ) ⊗ A X Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 116 and ψ : P gr( M k ( A )( d )) − → P gr( A ) (5.3) Y 7− → A k ( − d ) ⊗ M k ( A )( d ) Y . The functor φ induces a homomorphism f rom K gr i ( A ) t o K gr i ( M k ( A )( d )). By the graded vers ion of the Morita Theorems (see Prop osition 4.5.4 ), the functor ψ es- tablishes a natural equiv alence of catego ries, so it induces an isomorphism from K gr i ( M k ( A )( d )) to K gr i ( A ). F or X ∈ P gr( A ), ψ ◦ φ ( X ) ∼ = gr X k ( − d ). W e will use a similar argumen t to tha t of Prop osition 4.2.2 to show that X k ( − d ) ∼ = gr X k . Since ( d ) ∈ Γ ∗ M k ( R ) , there is r = ( r ij ) ∈ GL k ( R )[ d ]. Then there is a graded A -mo dule homomo r phism L r : X k − → X k ( − d ) ( x 1 , . . . , x k ) 7− → r ( x 1 , . . . , x k ) . Since r is in v ertible t here is an inv erse matrix t ∈ GL k ( R ), and there is an A -mo dule homomorphism L t : X k ( − d ) → X k whic h is an in v erse of L r . So L r is a gr a ded A -mo dule isomorphism. By the remarks after Theorem 2.3.5 , K i are functors whic h resp ect direct sums, so this induces a m ultiplication by k on the leve l of the K -gro ups. The exact functors ( 5.2 ) and ( 5.3 ) induce the following commutativ e dia gram: K gr i ( R ) / / =   K gr i ( A ) / / φ   CK gr i ( A ) / /   1 K gr i ( R ) / / η k   K gr i ( M k ( A )( d )) / / ψ ∼ =   CK gr i ( M k ( A )( d )) / / ρ   1 K gr i ( R ) / / K gr i ( A ) / / CK gr i ( A ) / / 1 where comp osition of the columns are η k , proving prop erty (2). A diagram c hase v eriﬁes that prop ert y (3) also holds.  Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 117 A similar pro of shows that ZK gr i is a lso a gra ded D -functor. W e now sho w that for a graded Azumay a algebra whic h is graded free o v er its cen tre, its graded K - theory is essen tially the same as the graded K - t heory o f its cen tre. Theorem 5.4.4. L et A b e a gr ade d Azumaya algebr a which is gr ade d fr e e over its c en tr e R of dimension n , such that A has a h omo gen e ous b asis with de gr e es ( δ 1 , . . . , δ n ) in Γ ∗ M n ( R ) . Then for any i ≥ 0 , K gr i ( A ) ⊗ Z [1 /n ] ∼ = K gr i ( R ) ⊗ Z [1 /n ] . Pro of. Prop o sition 5.4.3 sho ws that CK gr i (and in the same manner ZK gr i ) is a graded D -functor, and thus b y Theorem 5.4.2 CK gr i ( A ) and ZK gr i ( A ) are n 2 -torsion ab elian groups. T ensoring the exact sequence ( 5.1 ) b y Z [1 /n ], since CK gr i ( A ) ⊗ Z [1 /n ] and ZK gr i ( A ) ⊗ Z [1 /n ] v anish, the result follows .  It remains as a question when this result holds for a gr a ded Azumay a algebra of constan t rank. Question 5.4.5. Let Γ b e an ab elian group, R b e a commutativ e Γ-graded ring, and A b e a graded Azumay a algebra o v er its cen tre R of rank n . When is it true that for an y i ≥ 0, K gr i ( A ) ⊗ Z [1 /n ] ∼ = K gr i ( R ) ⊗ Z [1 /n ]? Remark 5.4.6. Using Example 5.3.2 , w e remark here t ha t t he g r aded Azumay a algebra H do es not satisfy the conditions of Theorem 5.4.4 . Supp ose H do es satisfy these conditio ns; that is, supp o se there is a homogeneous basis { a 1 , . . . , a 4 } for H o v er R , suc h that the elemen ts of the basis hav e degrees ( δ 1 , . . . , δ 4 ) in Γ ∗ M 4 ( R ) . So there exists a matrix r ∈ G L 4 ( R )[ d ]. Since Supp( R ) = 0, then as eac h row of r must con tain a non-zero elemen t, this implies δ i = 0 fo r eac h i . But this w ould imply that the supp ort o f H is also 0, whic h clearly is a con tradiction. So suc h a homogeneous basis for H do es not exist. Chapter 5. G r a ded K -Theory of Azuma y a Alg ebras 118 Corollary 5.4.7. L et A b e an Azumaya algebr a fr e e over its c entr e R of dim ension n . Then for any i ≥ 0 , K i ( A ) ⊗ Z [1 /n ] ∼ = K i ( R ) ⊗ Z [1 /n ] . Pro of. By taking Γ to b e the trivial group, this follows immediately from Theo- rem 5.4.4 .  Chapter 6 Additiv e Comm utators In 1905 W edderburn prov ed that a ﬁnite division ring is a ﬁeld. This is an example of a comm uta tivit y theorem; that is, it is a theorem whic h states certain conditions under whic h a giv en ring is comm utative . Since then, W edderburn’s result has motiv ated man y , more general comm utativity theorems (see [ 42 , § 13]). Both additiv e and m ultiplicativ e comm utators pla y an imp ortan t ro le in these theorems. In this c hapter w e consider additiv e comm utato rs in the setting of graded division algebras. W e b egin Section 6.1 with t wo results in v olving the supp ort of a graded division ring. W e then show that some comm utativit y theorems inv olving additiv e comm u- tators hold in the g raded setting. W e giv e a coun ter-example to sho w that one suc h comm utativit y theorem for m ultiplicativ e comm utators do es not hold. In Section 6.2 , w e sho w that in the setting of graded division algebras, the reduced trace exists and it is a graded map. In Section 6.3 , w e recall some results from the non-graded setting whic h will b e used in the ﬁnal section. W e end the c hapter by considering, in Section 6.4 , the quotien t division ring QD of a g r a ded division ring D . W e sho w ho w t he submo dule generated b y the additive commutators in QD relates to that of D (see Corollary 6.4.5 ). 119 Chapter 6. Additive Commutators 120 6.1 Homogeneo u s additiv e co mm utators Throughout this chapter, let Γ b e an ab elian group unless o therwise stated. W e recall f r o m Section 4.1 that the supp ort of a graded ring R = L γ ∈ Γ R γ is deﬁned to b e the set Supp( R ) = Γ R =  γ ∈ Γ : R γ 6 = { 0 }  . It follow s that a graded ring R is zero if a nd o nly if Supp( R ) = ∅ . Let D b e a Γ-graded division ring with cen tre F . Since Γ is an ab elian g roup, the cen tre of D is a graded subring of D (see Section 4.1 ). A homo gene o us additive c ommutator of D is deﬁned to b e an elemen t of the form ab − ba where a, b ∈ D h . Throughout this c hapter, w e will use the notation [ a, b ] = ab − ba and [ D , D ] is the graded submo dule of D generated b y all homogeneous additiv e commutators o f D . W e note that in this c hapter w e will consider the group Γ to b e an ab elian group, unless stated otherwise. This is the natural setting to consider homogeneous additiv e comm utators. If Γ is not ab elian, then a giv en homogeneous additive commutator ma y not b e a homog eneous elemen t in D . A gr ade d division algeb r a D is deﬁned to b e a graded division ring with cen tre F suc h that [ D : F ] < ∞ . Not e that since F is a graded ﬁeld, D has a ﬁnite homogeneous basis ov er F . A graded division algebra D o ve r its cen tre F is said to b e unr amiﬁe d if Γ D = Γ F and total ly r amiﬁe d if D 0 = F 0 . The following lemma considers the supp ort of [ D , D ]; the pro o f of part ( 2 ) is due to Hazrat. Lemma 6.1.1. L et D = L γ ∈ Γ D γ b e a gr ade d division a l g ebr a over its c entr e F . 1. If D is total ly r amiﬁe d, then ∅ 6 = Supp([ D , D ]) $ Γ D . 2. If D is not total ly r am iﬁe d, then Supp( D ) = Supp([ D , D ]) . Pro of. (1): Clearly ∅ 6 = Supp([ D , D ]) ⊆ Γ D . Since D 0 = F 0 = Z ( D ) ∩ D 0 w e ha v e D 0 ⊆ Z ( D ). Supp ose 0 ∈ Supp([ D , D ]). Then there is an elemen t P i ( x i y i − y i x i ) ∈ [ D , D ], with deg( x i ) + deg ( y i ) = 0 for all i . If x i y i − y i x i = 0 f or all i , then clearly Chapter 6. Additive Commutators 121 the sum is also zero. Thus there are non- zero ho mogeneous ele men ts x ∈ D γ , y ∈ D δ with 0 6 = xy − y x ∈ D 0 and γ + δ = 0 . Then ( xy − y x ) y − 1 6 = 0, as y − 1 ∈ D − δ \ 0 a nd xy − y x ∈ D 0 \ 0, so their pro duct is a non-zero homogeneous elemen t of degree − δ . Since ( xy − y x ) y − 1 = xy y − 1 − y xy − 1 = y − 1 y x − y xy − 1 , w e hav e y − 1 ( y x ) 6 = ( y x ) y − 1 ; that is y x / ∈ Z ( D ). Since y x ∈ D 0 , this contradicts the fact that D 0 = F 0 , so 0 / ∈ Supp([ D , D ]). (2): It is clear that Supp([ D , D ]) ⊆ Γ D . F or the rev erse containmen t, for γ ∈ Γ D w e will show that there is a n x ∈ D γ whic h do es no t comm ute with some y ∈ D δ for some δ ∈ Γ D . Supp ose not, then D γ ⊆ Z ( D ), so D γ = F γ . Let x ∈ D γ , d ∈ D 0 , y ∈ D δ b e arbitrary non- zero elemen ts. Then x ( dy ) = ( dy ) x = d ( y x ) = d ( xy ) = ( dx ) y = y ( dx ) = ( y d ) x = x ( y d ) . So for all d ∈ D 0 , y ∈ D δ w e ha v e x ( dy ) = x ( y d ). Since x is a non-zero homogeneous elemen t, it is in v ertible. This implies dy = y d , so D 0 = F 0 con tradicting the fa ct that D is not t o tally ramiﬁed. Then there is an x ∈ D γ whic h do es not comm ute with y ∈ D δ , so xy y − 1 − y − 1 xy 6 = 0 proving γ ∈ Supp([ D , D ]).  Example 6.1.2. Let H b e the real quaternion alg ebra. W e sa w in Example 4.2.3 (2) that H forms a Z 2 × Z 2 -graded division ring, where its ce ntre is R . Then Supp( H ) = Z 2 × Z 2 and Supp( R ) = (0 , 0). W e can sho w tha t Supp([ H , H ]) = { (1 , 0) , (0 , 1) , (1 , 1) } . Let D b e a g raded division r ing and let F b e a graded subﬁeld of D whic h is con tained in t he cen tre of D . W e kno w that F 0 = F ∩ D 0 is a ﬁeld and D 0 is a division ring. The group of inv ertible homogeneous elemen ts of D , denoted b y D h ∗ , is equal t o D h \ 0. Considering D as a graded F -mo dule, since F is a graded ﬁeld, there is a uniquely deﬁned dimension [ D : F ] b y Theorem 4.2.4 . Note that Γ F ⊆ Γ D , so Γ F is a normal subgroup of Γ D . Chapter 6. Additive Commutators 122 The prop osition b elo w has been prov en b y Hw a ng, W adsw ort h [ 38 , Prop. 2.2] for t w o gr a ded ﬁelds R ⊆ S with a torsion-free ab elian grade group. Prop osition 6.1.3. L et D b e a gr ade d divi s i on ring and l e t F b e a gr ade d subﬁeld of D whi ch is c ontaine d in the c entr e of D . Th e n [ D : F ] = [ D 0 : F 0 ] | Γ D : Γ F | . Pro of. Let { x i } i ∈ I b e a basis for D 0 o v er F 0 . Consider the cosets of Γ D o v er Γ F and tak e a transv ersal { δ j } j ∈ J for these cosets, where δ j ∈ Γ D . T ake { y j } j ∈ J ⊆ D h ∗ suc h that deg( y j ) = δ j for eac h j . W e will sho w that { x i y j } is a basis for D o v er F . Consider the map ψ : D h ∗ − → Γ D / Γ F d 7− → deg ( d ) + Γ F . This is a group homomorphism with k ernel D 0 F h ∗ , since for an y d ∈ k er ( ψ ) there is some f ∈ F h ∗ with d f − 1 ∈ D 0 . Let d ∈ D b e arbitrary . Then d = P γ ∈ Γ d γ where d γ ∈ D γ and ψ ( d γ ) = γ + Γ F = δ j + Γ F for some δ j in the transv ersal of Γ D o v er Γ F . Then there is some y j with deg ( y j ) = δ j and d γ y − 1 j ∈ k er( ψ ). So d γ y − 1 j = P k a k g k for g k ∈ F h ∗ and a k = P i r ( k ) i x i with r ( k ) i ∈ F 0 . It follow s that d can b e written as an F -linear com bination of the elemen ts of { x i y j } . T o sho w linear independence, supp ose P n i =1 r i x i y i = 0 f or r i ∈ F . Since the homogeneous comp onen ts of D are disjoint, w e can tak e a ho mogeneous comp o nent of this sum, sa y P m k =1 r k x k y k where deg ( r k x k y k ) = α . Then deg ( r k ) + deg( y k ) = α for a ll k , so a ll of the y k are the same. This implies that P k r k x k = 0, where all of the r k ha v e t he same degree. If r k = 0 for a ll k then w e are done. Otherwise, f or some r l 6 = 0, we ha v e P k ( r − 1 l r k ) x k = 0. Since { x i } forms a basis for D 0 o v er F 0 , this implies r k = 0 for all k .  W e include b elow a num b er of results in v olving homogeneous additiv e comm u- Chapter 6. Additive Commutators 123 tators. The pro ofs follow in exactly the same w ay as the pro ofs of the equiv alen t non-graded results (see [ 43 , § 13]). Lemma 6.1.4. L et D b e a gr ade d division rin g. If al l of the homo gene ous addi tive c ommutators of D ar e c entr al, then D is a gr a de d ﬁeld. Pro of. Let y ∈ D h b e an arbitrary homog eneous elemen t of D . Then b y assump- tion, y comm utes with all homogeneous additive comm utators, and it follows that y comm ut es with a ll (non-homog eneous) additiv e comm utators of D . W e will show y ∈ Z ( D ). Supp ose y / ∈ Z ( D ). T hen there exists x ∈ D h suc h that [ x, y ] 6 = 0. W e hav e [ x, xy ] = x [ x, y ] with [ x, xy ] and [ x, y ] non-zero. Since y comm utes with [ x, xy ] and [ x, y ], it follo ws that ( y x − xy )[ x, y ] = 0. As w e a ssumed [ x, y ] 6 = 0, it is a non- zero homogeneous elemen t of D , so is in v ertible. So y x − xy = 0; that is, [ x, y ] = 0 con tradicting our c hoice of x . It fo llows that D is a comm utative graded division ring, as required.  W e include an a lt ernativ e pro of to Lemma 6.1.4 in Section 6.4 . This a lt ernat iv e pro of uses the relation b etw een a gra ded division ring and its quotien t division ring com bined with the non-gr a ded result. Theorem 6.1.5. L e t D b e a gr ade d div i s ion ring with c entr e F . T hen the smal lest gr ade d division subring over F gener ate d by homo gene ous additive c ommutators is D . Pro of. Let E = L γ ∈ Γ E γ b e the smallest g r aded F -division subring con taining a ll homogeneous additiv e commutators. Then clearly w e hav e F ⊆ E ⊆ D . T o sho w D = E , it is suﬃcien t to sho w that D h ⊆ E h . Let x ∈ D h \ F h . Then there exists y ∈ D h suc h that xy 6 = y x . W e ha v e [ x, xy ] = x [ x, y ] with [ x, xy ] , [ x, y ] ∈ E h , since they are non-zero homogeneous additive comm utato rs. Then x = [ x, xy ] · [ x, y ] − 1 ∈ E h , as required.  Chapter 6. Additive Commutators 124 Prop osition 6.1.6. L et K ⊆ D b e gr ade d division rings, with [ D , K ] ⊆ K . If c har K 6 = 2 , then K ⊆ Z ( D ) . Pro of. First note that t he condition [ D , K ] ⊆ K is equiv alent to [ D h , K h ] ⊆ K h . Let a ∈ D h \ K h and c ∈ K h . W e will sho w ac = ca . W e ha v e [ a, [ a, c ]] + [ a 2 , c ] = 2 a [ a, c ] ∈ K . If [ a, c ] 6 = 0, then, since c har K 6 = 2 , this implies a ∈ K , contradicting our c hoice of a . Th us [ a, c ] = 0. No w let b, c ∈ K h . Consider a ∈ D h \ K h . Then a, ab ∈ D h \ K h and we hav e sho wn that [ a, c ] , [ ab, c ] = 0. Then [ b , c ] = a − 1 · [ ab, c ] = 0. It follo ws that K ⊆ Z ( D ).  The ab ov e results sho w that, in some asp ects, the b eha viour o f homogeneous additiv e commutators in graded division rings is similar to that of additiv e com- m utators in division rings. How ev er, this analogy seems to fail for multiplic ative comm utators of graded division rings. F or example, in the setting of division rings the Cartan-Bra uer-Hua theorem is the m ultiplicativ e v ersion of Prop osition 6.1.6 and its pro of inv olves m ultiplicativ e commutators. In the non-g r a ded setting, the Cartan-Brauer-Hua theorem states: Let D and K b e division r ing s, with K ⊆ D . Supp ose that K ∗ is a normal subgroup of D ∗ and K 6 = D . Then K ⊆ Z ( D ). This theorem do es not ho ld in the setting of gr a ded division rings, as is sho wn b y the follo wing counterex ample. Example 6.1.7. Let D b e a division ring a nd let R = D [ x, x − 1 ]. Then w e saw in Example 4.2.3 (1) that R is a graded division ring. Let K = R 0 . Then K ∗ is a normal subgroup of R ∗ and K 6 = R . Cho ose a, b ∈ D with ab 6 = ba . Then ax 0 ∈ K and bx n ∈ R , and w e hav e ( ax 0 )( bx n ) = abx n 6 = bax n = ( bx n )( ax 0 ). So ax 0 / ∈ Z ( R ), pro ving that K * Z ( R ). Consider a homogeneous m ultiplicativ e comm utator xy x − 1 y − 1 in a graded divi- sion ring D . Since x, y ∈ D h and deg( x − 1 ) = − deg( x ), we note that deg( xy x − 1 y − 1 ) = Chapter 6. Additive Commutators 125 0; that is, xy x − 1 y − 1 is in the zero-homo g eneous compo nent of D . This suggests that there are “ to o few” m ultiplicativ e comm utators to aﬀect the structure of the division ring. 6.2 Graded splitting ﬁeld s F or a graded division algebra D o v er its cen t r e F , w e will sho w that D is split b y an y graded maximal subﬁe ld L of D . This allo ws us to construct the reduced c haracteristic p o lynomial of D ov er F . Lemma 6.2.1 (Graded Sc h ur’s Lemma) . L et R b e a gr ade d ring and let M b e a gr ade d R -mo d ule. If M is gr ade d simple, then END R ( M ) is a gr ade d di v ision ring. Pro of. W e kno w f r o m pag e 76 that END R ( M ) is a graded ring . If f is a nonzero homogeneous endomorphism, then k er ( f ) 6 = M and im( f ) 6 = 0. Sin ce ker( f ) and im( f ) are b ot h graded submo dules of M , whic h is graded simple, it follo ws t hat k er( f ) = 0 and im( f ) = M . Th us f is a g raded isomorphism, and hence is in v ertible.  In the follo wing theorem, w e rewrite Rieﬀel’s pro of of W edderburn’s Theorem in the graded setting (see [ 39 , Prop. 1.3 (a)]). See [ 51 , Thm. 2.10.10] for a more general v ersion of the theorem. Theorem 6.2.2. L et A b e a gr ade d c entr al sim p le a lgebr a over a gr ade d ﬁeld R . Then ther e is a gr ade d division alge b r a E over R such that A ∼ = gr M n ( E )( d ) for some ( d ) = ( δ 1 , . . . , δ n ) ∈ Γ n . Pro of. T ake a minimal nonzero homogeneous rig h t ideal L o f A (whic h exists since [ A : R ] < ∞ ). Then L is a graded simple right A -mo dule, as it has no nonzero prop er graded right A -submo dules. Let E = End A ( L ), where End A ( L ) = END A ( L ) since L is ﬁnitely g enerated as a graded A - mo dule. Then E is a g r a ded division ring, by the graded Sc h ur’s Lemma, and [ E : R ] ≤ [End R ( L ) : R ] < ∞ . Then L is Chapter 6. Additive Commutators 126 a graded free left E - mo dule, with a homogeneous base, say b 1 , . . . , b n , of L o ve r E . Then by Theorem 4.4.2 , E nd E ( L ) ∼ = gr M n ( E )( d ), w here ( d ) = (deg( b 1 ) , . . . , deg( b n )). Consider the map i : A − → End E ( L ) a 7− → i ( a ) : L → L x 7→ xa W e will sho w that i is a graded R - algebra isomorphism. Firstly note that i ( a ) is an elemen t of End E ( L ), since for all e ∈ E , x ∈ L w e hav e ( i ( a ))( ex ) = ( ex ) a = ( e ( x )) a = e ( xa ) = e  ( i ( a ))( x )  . It can b e easily sho wn that i is a graded R -algebra homomorphism. Since A is graded simple, it follows that i is injectiv e. Note that i ( A ) contains the iden tit y elemen t of End E ( L ). Then to prov e surjectivity , it suﬃces to show that i ( A ) is a homog eneous righ t ideal of End E ( L ). F or y ∈ L , let L y : L → L ; x 7→ y x so that L y ∈ End A ( L ). Then f or f ∈ End E ( L ) h , x ∈ L w e hav e f ( y x ) = f ( L y ( x )) = L y ( f ( x )) = y ( f ( x )) . It fo llo ws t ha t ( i ( x ) · f )( y ) = ( f ◦ i ( x ))( y ) =  i ( f ( x ))  ( y ), whic h implies that i ( L ) is a right ideal o f End E ( L ). Also, i ( L ) = L γ ∈ Γ  i ( L ) ∩ (End E ( L )) γ  giving that the ideal is homogeneous. The t wo-sided homogeneous ideal of A generated b y L is ALA = AL , and since A is gra ded simple, AL = A . Thus i ( A ) = i ( AL ) = i ( A ) i ( L ) is a homogeneous righ t ideal o f End E ( L ), as required.  Let D b e a gr aded division algebra o v er its centre F and let L b e an y graded subﬁeld of D containing F . W e deﬁne a grading on L [ x ] as follo ws. Let θ ∈ Γ D and Chapter 6. Additive Commutators 127 let L [ x ] θ = M γ ∈ Γ L [ x ] γ , where L [ x ] γ = n X a i x i : a i ∈ L h , deg( a i ) + iθ = γ o . Then L [ x ] θ forms a graded ring, and x ∈ L [ x ] θ is homogeneous of degree θ . Let Z D ( L ) = { d ∈ D : dl = l d for all l ∈ L } denote the cen traliser of L in D . Then Z D ( L ) is a graded subring of D and it is a graded L -algebra. F or an y c ∈ Z D ( L ) h of degree θ , let L [ c ] =  f ( c ) : f ( x ) ∈ L [ x ] θ  . Then L [ c ] forms a graded ring with L ⊆ L [ c ], and we note that L [ c ] is comm u- tativ e since c ∈ Z D ( L ) h . The map L [ x ] θ → L [ c ]; f ( x ) 7→ f ( c ) is a graded ring homomorphism. F urther w e will sho w that L [ c ] is in fact a graded ﬁeld. Let a ∈ L [ c ] γ b e a non-zero elemen t, and c onsider the γ -shifted L -mo dule L [ c ]( γ ). The map L a : L [ c ] → L [ c ]( γ ); l 7→ al is a graded L -mo dule homomorphism, whic h is injectiv e since a is in v ertible in D . Then dim L (im( L a )) = dim L ( L [ c ]). W e will sho w that dim L ( L [ c ]) < ∞ . Since c ∈ D and [ D : F ] < ∞ , w e ha v e tha t c is algebraic ov er F , a nd th us is algebraic o ve r L . So it ha s a minimal p olynomial h ( x ) = l 0 + l 1 x + · · · + l k x k , and the set { 1 , c, c 2 , · · · , c k − 1 } gene rat es L [ c ] o v er L . If they are not all linearly independen t, w e can write 1 as an L -linear com bination o f the others and remo v e it from the set, lea ving a set whic h still generates L [ c ] o v er L . Repeating this pro cess will giv e a linearly indep enden t spanning set for L [ c ] o v er L . So dim L ( L [ c ]) < ∞ , and L a is surjectiv e b y dimension count. Then there is a graded L -mo dule homomorphism ψ whic h is the in v erse o f L a , a nd ψ (1 L ) is the in v erse of a . Since L [ c ] is a lso comm utativ e, it is a g r aded ﬁeld. The follow ing r esult is in [ 20 , p. 40] in the non-graded setting. Theorem 6.2.3. L et D b e a gr ade d division algebr a over a gr ade d ﬁ eld F , and let L b e a gr ade d subﬁeld of D . T h en L is a gr ade d m aximal subﬁeld if and only if Chapter 6. Additive Commutators 128 Z D ( L ) = L . Pro of. If Z D ( L ) = L , then for any graded subﬁeld L ′ of D con taining L w e ha v e L ′ ⊆ Z D ( L ). So L is a graded maximal subﬁeld. Con v ersely , assume L is graded maximal. Then for an y homogeneous c ∈ Z D ( L ), L [ c ] f orms a graded ﬁeld. By the maximality of L , we mus t ha v e L [ c ] = L and so c ∈ L . Then it follo ws that Z D ( L ) = L .  Corollary 6.2.4. L et D b e a gr ade d div i s ion algebr a with c entr e F a nd let L b e a gr ade d m a ximal subﬁeld of D . Then D ⊗ F L ∼ = gr M n ( L )( d ) for some n ∈ N and some d = ( δ 1 , . . . , δ n ) ∈ Γ n . Pro of. As graded F -mo dules, w e ha v e D ⊗ F L ∼ = gr L ⊗ F D . Since D is a gra ded division algebra o v er the graded ﬁeld F , and L is gra ded simple, Theorems 4.3.1 , 4.3.2 giv e that L ⊗ F D is graded cen tral simple ov er Z ( L ) = L . W e hav e that D is a graded simple righ t mo dule ov er L ⊗ F D , with the righ t action x ( l ⊗ d ) = lxd for d, x ∈ D , l ∈ L . Then b y Theorem 6.2.2 , E := End L ⊗ F D ( D ) is a gr aded division alg ebra o v er L , such that L ⊗ F D ∼ = gr M n ( E )( d ) for some d = ( δ 1 , . . . , δ n ) ∈ Γ n . F rom Theorem 6.2.3 , Z D ( L ) = L , so it remains to sho w Z D ( L ) ∼ = gr E . Deﬁne ψ : Z D ( L ) → End L ⊗ F D ( D ) d 7→ ψ ( d ) : D → D x 7→ dx It can b e easily sho wn that ψ is a graded L -alg ebra homomorphism, w hich is injectiv e since L is g raded simple as a graded ring. Let f ∈  End L ⊗ F D ( D )  γ b e a homogeneous map. Then f (1) ∈ D γ and since f (1) ℓ = ℓf (1) for all ℓ ∈ L h , w e ha v e f (1) ∈ Z D ( L ). F or x ∈ D , f ( x ) = f (1 · (1 ⊗ x )) = f (1)( 1 ⊗ x ) = f ( 1 ) x =  ψ ( f (1))  ( x ) . Chapter 6. Additive Commutators 129 So f = ψ ( f (1)), proving that ψ is surjectiv e. It follo ws that M n ( E )( d ) ∼ = gr M n ( L )( d ), completing the pro of.  Supp ose D is a graded division algebra ov er a g r aded ﬁeld F . W e will show that there exists a graded maximal subﬁeld of D . F or any a ∈ D h \ F h , w e hav e sho wn ab ov e that F [ a ] is a graded ﬁeld, with F ⊆ F [ a ] ⊆ D . Consider X = { L : L is a graded subﬁeld of D with F $ L } . This is a non-empt y set since F [ a ] ∈ X a nd it is partially o r dered with inclusion. Ev ery chain L 1 ⊆ L 2 ⊆ . . . in X has an upp er b ound S L i ∈ X . By Zo r n’s Lemma, X has a maximal elemen t, so there is a graded maximal subﬁeld of D . Then Corollary 6.2.4 sho ws that a graded max imal subﬁeld L of D splits D ; that is, j : D ⊗ L ∼ = gr M n ( L )( d ). As in the non-graded setting, fo r an elemen t d ∈ D w e deﬁne the reduced c haracteristic p olynomial as c har D /F ( d, x ) = det  xI n − j ( d ⊗ 1)  = x n − T rd D ( d ) x n − 1 + · · · + ( − 1) n Nrd D ( d ) , where T rd D ( d ) = trace  j ( d ⊗ 1)  is the reduced trace of d and Nrd D ( d ) = det  j ( d ⊗ 1)  is the reduced norm. Since L is a graded mo dule o v er a gra ded ﬁeld F , b y Prop osition 4.2.4 , it is graded free and therefore fr ee o v er F . It follow s that L is faithfully ﬂat ov er F , where F is a ring a nd D and L are F -algebras, so w e can apply [ 41 , Lemma I I I.1.2.1]. This sho ws that the reduced characteristic p olynomial lies in F [ x ] and it is indep enden t of the c hoice of j and L . W e note that the reduced norm and reduced trace satisfy the follo wing prop erties. Chapter 6. Additive Commutators 130 Corollary 6.2.5. R e duc e d norm and tr ac e s a tisfy the fol lowing rules: Nrd D ( ab ) = Nrd D ( a )Nrd D ( b ) T rd D ( a + b ) = T rd D ( a ) + T rd D ( b ) Nrd D ( r a ) = r n Nrd D ( a ) T rd D ( r a ) = r T rd D ( a ) T rd D ( ab ) = T rd D ( ba ) for al l a, b ∈ D and r ∈ F . Pro of. This follow s immediately from the prop erties of determinan t and trace in a matrix.  It follo ws fro m the ab ov e corollary that T rd D : D → F is an F - mo dule homo- morphism. Since a ll three maps D → D ⊗ F L → M n ( L )( d ) trace − → F are gra ded maps, w e ha v e that T rd D is a graded F -mo dule homomorphism. Note that a graded division a lgebra D with cen tre F is an Azuma y a alg ebra b y Theorem 4.3.3 . Since t he dimension of an Azuma y a algebra is a square, it follo ws that [ D : F ] is also a square num b er. Prop osition 6.2.6. L e t D b e a gr ade d di v i s ion algebr a ov e r its c entr e F . Then T rd D : D → F is surje ctive. Pro of. Supp ose T rd D is not surjectiv e. Since im (T rd D ) is a graded mo dule o v er the g raded ﬁeld F with dim F (im(T rd D )) ≤ dim F ( F ) = 1, then dim (im(T rd D )) = 0 and so T rd D is the zero map. Let { x 1 , . . . , x n 2 } b e a homogeneous basis for D o v er F and let L b e a graded maximal subﬁeld of D . Then by Corollary 6.2.4 , f : D ⊗ F L ∼ = gr M n ( L )( d ) for some ( d ) ∈ Γ n and it is kno wn that { x 1 ⊗ 1 , . . . , x n 2 ⊗ 1 } forms a homogeneous basis for D ⊗ F L ov er L . Since f is a graded isomorphism, { f ( x i ⊗ 1) : 1 ≤ i ≤ n 2 } forms a homogeneous basis of M n ( L )( d ) ov er L . By deﬁnition T rd D ( d i ) = tr( f ( d i ⊗ 1 ) ), whic h equals zero since T r d D is the zero map. That is, the tra ce is the zero function on M n ( L )( d ), whic h is clearly a con tradiction.  Chapter 6. Additive Commutators 131 6.3 Some r e sults in the non-graded setting W e recall here some results from the non-graded setting, whic h will b e used in t he next section. Lemma 6.3.1. L et R b e a division ring. If al l of the additive c ommutators of R a r e c entr a l, then R i s a ﬁeld. Pro of. See [ 43 , Cor. 13.5].  Theorem 6.3.2. L et D b e a d ivision algebr a over its c entr e F of index n . Then for a ∈ D , T rd D ( a ) = na + d a wher e d a ∈ [ D , D ] . Pro of. Let a ∈ D with minimal p olynomial f ( x ) ∈ F [ x ] o f degree m . Then by [ 56 , p. 124, Ex. 9.1], w e hav e f ( x ) n/m = x n − T rd D ( a ) x n − 1 + · · · + ( − 1) n Nrd D ( a ) . where the right hand side of this equalit y is the reduced characteris tic po lyno- mial of a . W edderburn’s F actorisation Theorem [ 42 , Thm. 1 6.9] sa ys f ( x ) = ( x − d 1 ad − 1 1 ) · · · ( x − d m ad − 1 m ) for d 1 , . . . , d m ∈ D . Com bining these, we hav e T rd D ( a ) = n m  d 1 ad − 1 1 + · · · + d m ad − 1 m  = n m  ma + ( d 1 ad − 1 1 − ad − 1 1 d 1 ) + · · · + ( d m ad − 1 m − ad − 1 m d m )  = na + d a where d a ∈ [ D , D ] , as required.  Let R b e a comm utativ e No etherian ring . The dimension of the maximal ideal space of R is deﬁned to b e the suprem um on the lengths of prop erly descending c hains of irreducible closed sets. Theorem 6.3.3. L et R b e a c ommutative No etherian ring and let A b e an Azumaya algebr a over R . Then every elem ent of A of r e duc e d tr ac e zer o is a sum of at mos t Chapter 6. Additive Commutators 132 2 d + 2 additive c ommutators, wher e d is the d i m ension of the maxim a l ide al sp ac e of R . Pro of. See [ 59 , Thm. 5.3.1].  Corollary 6.3.4. L et D b e a gr ade d division algebr a o ver its c entr e F , which is No etherian as a ring. Then k er(T r d D ) = [ D , D ] . Pro of. F o r a n y xy − y x ∈ [ D , D ], w e hav e T rd D ( xy − y x ) = 0 b y Corollary 6.2.5 . The rev erse con tainmen t follo ws immediately fr o m the ab ov e theorem, s ince b y The- orem 4.3.3 , D is an Azuma y a algebra o v er F .  Remark 6.3.5. Let D b e a graded division alg ebra ov er its centre F , whic h is No etherian as a ring. Since k er(T rd D ) = [ D , D ] b y Corollar y 6.3.4 and T rd D is surjec- tiv e b y Prop osition 6.2 .6 , the First Is omor phism Theorem says that D / [ D , D ] ∼ = gr F as graded F - mo dules. So dim F ( D / [ D , D ]) = dim F F = 1. By Prop osition 4.2.6 , dim F ([ D , D ]) + 1 = dim F ( D ) < ∞ . W e recall here the deﬁnitions of a totally ordered group and a torsion-free g roup. Let (Γ , +) b e a group. A partial o r der is a binary relation ≤ on Γ whic h is reﬂexiv e, an tisymmetric and transitiv e. The order relation is translation in v ariant if f or all a, b, c ∈ Γ, a ≤ b implies a + c ≤ b + c and c + a ≤ c + b . A partially ordered group is a group Γ equipp ed with a partial order ≤ whic h is translation inv ariant. If Γ has a partial order, then tw o distinct elemen ts a, b ∈ Γ a r e said to b e comparable if a ≤ b o r b ≤ a . If Γ is a partially ordered group in whic h eve ry tw o elemen ts of Γ are comparable, then Γ is called a total ly or d er e d gr oup . F or a group Γ, a n elemen t a of Γ is called a torsion elemen t there is a p ositiv e in teger n such t hat a n = e . If the only torsion elemen t is the iden tit y elemen t , then the group Γ is called torsion-fr e e . By [ 44 ], an ab elian group can b e equipp ed with a total order if and only if it is torsion-free. Chapter 6. Additive Commutators 133 6.4 Quotien t divi s ion rin g s Throughout this section, Γ is a torsion-fr ee ab elian group and all g r aded ob jects a re Γ-graded. In this setting, graded division rings ha v e no zero divisors, and similarly for graded ﬁelds. This follo ws since we can choose a tota l order for Γ. So for a g raded division ring D = L γ ∈ Γ D γ and t w o non-zero elemen t s a, b ∈ D , w e can write a = a γ + terms of higher degree and b = b δ + terms of higher degree . Then ab = a γ b δ + terms of higher degree, so we hav e that ab is non-zero. Th us the group of units of D is D ∗ = D h \ 0. Similarly , a graded ﬁeld F is an integral domain with group o f units F ∗ = F h \ 0. This allow s us to construct QF = ( F \ 0) − 1 F , the quotient ﬁeld o f F , whic h is clearly a ﬁeld and an F -mo dule. F o r a gra ded division algebra D with cen tre F , w e deﬁne the quotient division ring of D t o b e QD = QF ⊗ F D . W e observ e some prop erties of QD in the pro p osition b elow, including in part (5) that it is a division ring. Prop osition 6.4.1. L e t D b e a gr a d e d division algebr a with c entr e F , and let QF and QD b e as deﬁne d ab o v e . Then the fol lowing p r op erties hold : 1. QD is an alg ebr a over QF ; 2. D → QD ; d 7→ 1 ⊗ d is in je ctive; 3. QD has no zer o divisors; 4. [ QD : QF ] = [ D : F ] ; 5. QD is a divi s i on ring ; 6. QD ∼ = ( F \ 0) − 1 D ; ( f 1 /f 2 ) ⊗ d 7→ ( f 1 d ) /f 2 ; 7. The e lements o f QD c an b e written as d/f , d ∈ D, f ∈ F . Chapter 6. Additive Commutators 134 Pro of. (1): This follow s easily . (2): T o sho w D → QD is inj ective , w e ﬁrst sho w that F → QF is injectiv e. Consider F → QF ; f 7→ f / 1. If f / 1 = 0, there is s ∈ F \ 0 with sf = 0. Since F is an in tegral do ma in, it follo ws that f = 0 and the map is injective . As D is graded free o v er F , and thus ﬂat o v er F , the required r esult follo ws. (3): Let x ∈ QF ⊗ D , sa y x = P ( f i /f ′ i ) ⊗ d i . Then x = ( f 1 /f ′ 1 ) ⊗ d 1 + · · · + ( f k /f ′ k ) ⊗ d k = (1 /f ) ⊗ f 1 f ′ 2 · · · f ′ k d 1 + · · · + (1 /f ) ⊗ f k f ′ 1 · · · f ′ k − 1 d k = (1 /f ) ⊗ d, where f = f ′ 1 f ′ 2 · · · f ′ k and d ∈ D . So f o r any arbitrary elemen t x of QF ⊗ D , w e can write x = (1 /f ) ⊗ d , for f ∈ F , d ∈ D . Now let x = (1 /f ) ⊗ d a nd y = (1 /f ′ ) ⊗ d ′ b e arbitrary elemen ts of QF ⊗ D with xy = 0. Then 1 / ( f f ′ ) ⊗ dd ′ = 0, so ( f f ′ ⊗ 1 )  1 f f ′ ⊗ d d ′  = 0 . Th us 1 ⊗ dd ′ = 0 and since D → QF ⊗ D is injectiv e, w e ha v e dd ′ = 0. As D has no zero divisors, it follows that d = 0 or d ′ = 0; that is, x = 0 or y = 0 as required. (4): Since D is free o v er F and QD = QF ⊗ F D , w e hav e [ D ⊗ F QF : F ⊗ F QF ] = [ D : F ]; that is, [ QD : QF ] = [ D : F ]. (5): W e will sho w that any domain whic h is ﬁnite dimensional ov er a ﬁeld is a division ring. Since QF is a ﬁeld, using (3) and (4), it f o llo ws that QD is a division ring. Supp ose A is a domain, F is a ﬁeld, Z ( A ) = F and [ A : F ] = n < ∞ . Let a ∈ A \ 0 and consider 1 , a, a 2 , . . . , a n . Then t here are r i ∈ F , not all zero, with r 0 + r 1 a + · · · + r n a n = 0. If r j is the ﬁrst non-zero elemen t of { r 0 , . . . , r n } , then Chapter 6. Additive Commutators 135 r j + r j +1 a + · · · + r n a n − j = 0. As a 6 = 0 and F is a ﬁeld, w e hav e a ( r j +1 + · · · + r n a n − j − 1 )( − r j ) − 1 = 1, so a is in ve rtible. (6): F ollows immediately from [ 48 , Prop. 6.55]. Note that the y are isomorphic as QF -algebras. (7): W e know from part (3) that if x ∈ QD , then we can write x = (1 /f ) ⊗ d . F rom the isomorphism in (6 ), QF ⊗ D ∼ = ( F \ 0) − 1 D ; ( 1 /f ) ⊗ d 7→ d/f . So w e can consider the elemen t s o f QD as d/f for d ∈ D , f ∈ F \ 0.  Note that w e hav e QF ⊗ F D ∼ = D ⊗ F QF as QF -alg ebras, so w e will use the terms in t erchangeably . W e include here an alternativ e pro of of Lemma 6.1.4 , due to Hazrat, whic h follow s from the non- graded result b y using the quotien t division ring. Alternativ e pro of of Lemma 6.1.4 . Let y ∈ D h b e an elemen t whic h commutes with homogeneous additiv e comm utators of D . Then it follo ws that y comm utes with all (non-homogeneous) comm utators of D . Consider [ x 1 , x 2 ] where x 1 , x 2 ∈ QD , with x 1 = d 1 /f 1 and x 2 = d 2 /f 2 for d 1 , d 2 ∈ D , f 1 , f 2 ∈ F . Then y [ x 1 , x 2 ] = y ([ d 1 , d 2 ] /f 1 f 2 ) = y [ d 1 , d 2 ] /f 1 f 2 = [ d 1 , d 2 ] y /f 1 f 2 = ([ d 1 , d 2 ] /f 1 f 2 ) y = [ x 1 , x 2 ] y . So y comm utes with all comm utato r s of QD , a division ring. Using Lemma 6.3.1 , y ∈ QF . W e will sho w that D h ∩ QF ⊆ F h . If x ∈ D h ∩ QF , then x = d/ 1 = f ′ /f f or d ∈ D h , f , f ′ ∈ F with f 6 = 0. So there exis ts t ∈ F \ 0 with tf d = tf ′ , so that d = f − 1 f ′ . Th us x = ( f − 1 f ′ ) / 1 ∈ F with deg( d ) = deg( f − 1 f ′ ); that is, x ∈ F h . This pro v es that y ∈ F h . It follows that all elemen ts of D are comm utative, completing the pro of.  Prop osition 6.4.2. L et D b e a gr ade d division algebr a o ver its c entr e F . Then fo r a ∈ D h , the r e duc e d char acteristic p olynom ial of a with r esp e c t to D over F c oincides with the r e duc e d char acteristic p olynomial of a ⊗ 1 with r esp e ct to QD over QF . Pro of. Let L b e a splitting ﬁeld of QD ov er QF , which exists as QD is a ﬁnite dimensional division algebra ov er QF . So i : QD ⊗ QF L ∼ = M n ( L ), and therefore L Chapter 6. Additive Commutators 136 is a splitting ﬁeld of D o v er F , since j : D ⊗ F L ∼ = D ⊗ QF QF ⊗ F L ∼ = M n ( L ) . Then for a ∈ D , c har QD /QF ( a ⊗ 1) = det  xI n − i (( a ⊗ 1 QF ) ⊗ 1 L )  = de t  xI n − j ( a ⊗ 1 L )  = c har D /F ( a ) . So in particular, w e hav e T rd QD ( a ⊗ 1) = T rd D ( a ) and Nrd QD ( a ⊗ 1) = Nrd D ( a ).  Prop osition 6.4.3. L e t D b e a gr ade d division algebr a over i ts c entr e F , which is No etherian as a ring. Then [ D , D ] = [ QD , QD ] ∩ D . Pro of. W e observ ed in Remark 6.3.5 that dim F ([ D , D ]) + 1 = dim F ( D ) < ∞ . F or an y P i x i y i − y i x i ∈ [ D , D ], as D → QD is injectiv e, 1 ⊗ X i x i y i − y i x i = X i (1 ⊗ x i )(1 ⊗ y i ) − (1 ⊗ y i )(1 ⊗ x i ) ∈ [ QD , QD ] ∩ D . So w e hav e [ D , D ] ⊆ [ QD, QD ] ∩ D ⊆ D . Here D * [ QD , QD ], so [ QD , QD ] ∩ D 6 = D . Th us [ D , D ] = [ QD , QD ] ∩ D .  Corollary 6.4.4. L et D b e a gr a d e d divisio n a lgebr a with c entr e F of index n , wher e F is No etherian as a ring. T hen for e ach a ∈ D , T rd D ( a ) = na + d a for some d a ∈ [ D , D ] . Pro of. Let a ∈ D , where T rd QD ( a ⊗ 1) = T rd D ( a ) b y Prop osition 6.4.2 . Since QD is a division ring, by Theorem 6 .3 .2 , T rd D ( a ) = n ( a ⊗ 1) + c where c ∈ [ QD , QD ]. W e kno w T rd D ( a ) ∈ F . Since D → QD is injectiv e, w e can consider n ( a ⊗ 1) as an elemen t of D . So c = T rd D ( a ) − na ∈ [ QD , QD ] ∩ D = [ D , D ] b y Prop osition 6.4.3 , as required.  Chapter 6. Additive Commutators 137 The pro of of the ab o ve corollary sho ws that using the quotient division ring and the result in the non-graded setting, the gr aded result follow s immediately . W e note that this pro of only holds for division algebras with a torsion-free grade group. Corollary 6.4.5. L et D b e a gr ade d division algebr a o ver its c entr e F , which is No etherian as a ring. Then D [ D , D ] ⊗ F QF ∼ = QD [ QD , QD ] . Pro of. F ro m t he a b ov e results w e hav e D / [ D , D ] ∼ = gr F as graded F -mo dules. 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Math., 213 (1963/1 964), 187–199. 63 , 77 Index ab elianisation, 4 additiv e commutator, 115 algebra Azuma y a, 23 , 29 cen tral, 27 classically separable, 22 en v eloping, 17 ´ etale, 27 ﬁnitely presen ted, 2 7 separable, 17 an ti-homomor phism, 32 Azuma y a algebra, 23 , 29 cen tral algebra, 27 classically separable algebra, 22 comm utator subgroup, 4 constan t rank, 57 crossed pro duct, 62 D -functor, 53 Dieudonn ´ e determinan t, 4 0 elemen ta r y matrix, 38 en v eloping algebra, 17 ´ etale alg ebra, 27 exact functor, 11 faithful functor, 28 faithful mo dule, 12 faithfully exact functor, 11 faithfully ﬂat mo dule, 14 faithfully pro jectiv e mo dule, 14 ﬁeld extension separable, 22 ﬁnitely presen ted algebra, 2 7 ﬂat mo dule, 12 functor exact, 11 faithful, 28 faithfully exact, 11 , 12 full, 28 fully faithful, 28 general linear group, 38 generator mo dule, 28 go o d grading, 88 graded algebra, 72 graded cen tra l simple, 73 graded Azuma ya algebra, 91 graded cen tra l simple algebra, 73 graded D -functor, 107 graded division algebra, 115 145 Index 146 graded division ring, 69 graded ﬁeld, 73 graded ideal, 63 graded matrix ring, 77 graded mo dule, 65 graded faithfully pro j ective , 89 graded free, 65 graded pro jectiv e, 89 graded simple, 65 graded mo dule homomorphism, 6 5 graded Morita equiv alence, 92 graded ring, 61 graded simple, 63 graded ring homomorphism, 64 graded submo dule, 65 group torsion-free, 126 totally ordered, 126 Ho c hsc hild homology mo dule, 59 homogeneous additiv e comm utator, 115 homogeneous elemen t, 61 homogeneous ideal, 63 in v olution, 32 Jacobson radical, 22 lo cal ring, 57 mo dule faithful, 12 faithfully ﬂat, 14 faithfully pro jectiv e, 14 ﬂat, 12 generator, 28 pro jectiv e, 1 2 pro jectiv e g enerato r , 28 Morita equiv a lence, 37 principal ideal domain, 38 progenerator mo dule, 28 pro jectiv e g enerato r mo dule, 28 pro jectiv e mo dule, 12 prop er maximally cen tral, 33 quotien t division ring, 12 7 quotien t ﬁeld, 127 ring lo cal, 57 semi-lo cal, 40 semi-simple, 37 semi-lo cal ring, 40 semi-simple ring, 37 separabilit y idemp otent, 18 separable algebra, 17 separable ﬁeld extension, 22 Stein b erg gro up, 41 strongly graded ring, 62 supp ort, 61 torsion group, 44 Index 147 torsion-free group, 126 totally ordered group, 126 totally ramiﬁed, 115 trace ideal, 28 unramiﬁed, 115

K-Theory of Azumaya Algebras

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