Unitary SK1 of graded and valued division algebras, I

UNIT AR Y SK 1 OF GRADED AND V ALUED DIVISIO N ALGEBRAS, I R. HAZRA T AND A. R. W AD SWOR TH Abstra ct. The reduced unitary W h itehead group SK 1 of a graded division algebra equipp ed with a unitary inv olution (i.e., an inv olution of t he second kind ) and graded by a torsion-free ab elian group is studied. It is sho wn that calculations in the graded setting are much simpler than their nongraded counterparts. T he bridge to the non- graded case is established by pro ving that th e unitary SK 1 of a tame va lued division algebra wih a unitary inv olution o ver a henselian ﬁeld coincides with th e unitary S K 1 of its associated graded division algebra. As a consequen ce, th e graded approach allow s us not only to reco ver results a va ilable in the literature with su b stanti ally easier pro ofs, bu t also to calculate the u nitary SK 1 for much wi d er classes of division algebras over hen selian ﬁelds. 1. Introduction Motiv a ted b y Platono v’s striking work on the reduced Whitehead group SK 1 ( D ) of v alued d ivision alge- bras D , see [ P 2 , P 4 ], V. Y anchevski ˘ ı, considered the unitary analogue, SK 1 ( D , τ ), for a division algebra D with unitary (i.e., second kin d) in v olution τ , s ee [ Y 1 , Y 2 , Y 3 , Y 4 ]. W orking with d ivision algebras o v er a ﬁeld with henselian discrete (rank 1) v aluation wh ose residu e ﬁeld also con tains a henselian d iscrete v alua- tion, and carrying out f ormidable tec hnical calculatio ns , he pro d uced remark able analogues t o Platono v’s results. By relat in g SK 1 ( D , τ ) to data ov er the residue algebra, he s ho we d not only that SK 1 ( D , τ ) could b e non trivial but that it could b e an y ﬁn ite ab elian group, and he ga ve a form ula in the bicyclic case expressing SK 1 ( D , τ ) as a quotient of r elativ e Brauer groups. Over the y ears since then sev eral appr oac hes ha ve b een giv en to und erstanding and calculating the (non unitary) group SK 1 using diﬀerent metho ds, notably b y Ers ho v [ E ], Suslin [ S 1 , S 2 ], Merkurjev and Rost [ Mer ] (F or surv eys on the group SK 1 , see [ P 4 ], [ G ], [ Mer ] or [ W 2 , § 6].) Ho wev er, ev en after the passage of some 30 yea r s , there do es not seem to hav e b een an y impro v ement in calculating SK 1 in the unitary setting. Th is may b e due in part to the co mp lexit y of the formulas in Y anchevski ˘ ı’s w ork, and the diﬃculty in follo wing some of his argum en ts. This pap er is a sequel to [ HaW ] where the red uced Whitehead group SK 1 for a grad ed division algebra w as studied. Here we consider the reduced u nitary Whitehead group of a graded division algebra with unitary graded inv olution. As in our pr evious w ork, we will see that the graded calculus is muc h easier and more transparent than the n on-graded one. W e calc u late the unitary SK 1 in seve r al imp ortan t cases. W e also sh o w ho w this enables one to ca lculate the unitary SK 1 of a tame division algebra ov er a henselian ﬁeld, b y passage to the asso ciated graded division algebra. The graded approac h allo ws us not only to reco v er most of Y anc hevski ˘ ı’s results in [ Y 2 , Y 3 , Y 4 ], with v ery substan tially simpliﬁed pr o ofs, but also extend them to arbitrary v alue group s and to calc u late the un itary SK 1 for wider classes of division algebras. There is a signiﬁcant simpliﬁcation gained by considering arbitrary v alue groups fr om the outset, rather than tow ers of discrete v aluations. But the greatest gain comes from passage to the graded setting, where the reduction to arithmetic considerations in the degree 0 division s ubring is quic ke r and more transparent. W e b rieﬂy describ e our principal results. Let E b e a graded d ivision algebra, w ith torsion f r ee ab elian grade group Γ E , and let τ b e a unitary graded inv olution on E . “Unitary” means that the act ion of τ on The ﬁrst author ac knowledges the supp ort of EPSRC ﬁrst grant sc heme EP/D0369 5X/1. The second author would lik e to thank the ﬁrst author and Queen’s Un ivers ity , Belfast for their hospitalit y while th e researc h for this pap er was carried out. 1 2 R. H AZRA T AND A. R. W A D SW OR TH the cen ter T = Z ( E ) is n on trivial (see § 2.3 ). Th e r e duc e d unitary Whitehe ad gr oup for τ on E is deﬁned as SK 1 ( E , τ ) =  a ∈ E ∗ | Nrd E ( a 1 − τ ) = 1  a ∈ E ∗ | a 1 − τ = 1  , where Nrd E is the r educed norm map Nrd E : E ∗ → T ∗ (see [ HaW , § 3]). Here, a 1 − τ means a τ ( a ) − 1 . Let R = T τ = { t ∈ T | τ ( t ) = t } $ T (see § 2.3 ). Let E 0 b e the s ubring of homoge n eous elemen ts of degree 0 in E ; lik ewise for T 0 and R 0 . F or an in v olution ρ on E 0 , S ρ ( E 0 ) d enotes { a ∈ E 0 | ρ ( a ) = a } and Σ ρ ( E 0 ) = h S ρ ( E 0 ) ∩ E ∗ 0 i . Let n b e the index of E , an d e the exp onent of the group Γ E / Γ T . Since [ T : R ] = 2, there are just t wo p ossible case s: either (i) T is unramiﬁed o ver R , i.e., Γ T = Γ R ; or (ii) T is totally ramiﬁed o v er R , i.e., | Γ T : Γ R | = 2 . W e will prov e t h e follo w ing f ormulas for the unitary SK 1 : (i) Sup p ose T / R is unr amiﬁed: • If E / T is un r amiﬁed, th en SK 1 ( E , τ ) ∼ = SK 1 ( E 0 , τ | E 0 ) (Prop. 4.10 ). • If E / T is totally ramiﬁed , then (Th. 5.1 ): SK 1 ( E , τ ) ∼ =  a ∈ T ∗ 0 | a n ∈ R ∗ 0 }  { a ∈ T ∗ 0 | a e ∈ R ∗ 0 } ∼ =  ω ∈ µ n ( T 0 ) | τ ( ω ) = ω − 1  µ e . • If Γ E / Γ T is cyclic, and σ is a generator of Ga l ( Z ( E 0 ) /T 0 ), then (Prop. 4.13 ): ◦ SK 1 ( E , τ ) ∼ = { a ∈ E ∗ 0 | N Z ( E 0 ) / T 0 (Nrd E 0 ( a )) ∈ R 0 }  Σ τ ( E 0 ) · Σ στ ( E 0 )  . ◦ If E 0 is a ﬁeld, then SK 1 ( E , τ ) = 1. • If E has a maximal graded subﬁeld M unr amiﬁed o ver T and another maximal graded subﬁ eld L totally ramiﬁed o ver T , with τ ( L ) = L , th en E is semiramiﬁed and (C or. 4.11 ) SK 1 ( E , τ ) ∼ =  a ∈ E 0 | N E 0 / T 0 ( a ) ∈ R 0   Q h ∈ Gal( E 0 / T 0 ) E ∗ hτ 0 . (ii) If T / R is totally ramiﬁed, then SK 1 ( E , τ ) = 1 (Prop. 4.5 ). The bridge b et w een the graded and the non-graded henselian setting is esta b lish ed b y Th. 3.5 , whic h sho ws that for a tame division algebra D o v er a henselian v alued ﬁeld with a u nitary inv olution τ , SK 1 ( D , τ ) ∼ = SK 1 ( gr ( D ) , e τ ) where gr ( D ) is the graded division algebra asso ciated to D b y th e v aluation, and e τ is the graded inv olution on gr ( D ) indu ced by τ (see § 3 ). Th u s, eac h of the results listed ab o ve for graded division algebras yields analogous formulas for v alued division algebras ov er a henselian ﬁeld, as illustrated in Ex amp le 5.3 and Th. 5.4 . This reco v ers existing formulas, which w ere primarily for the case with v alue group Z or Z × Z , but with easier and more transparent pro ofs than those in the existing literature. Additionally , our r esu lts apply for an y v alue group s whateve r . The esp ecially simple case wh ere E / T is totally r amiﬁ ed and T / R is unr amiﬁ ed is entirely new. In the sequel to this p ap er [ W 3 ], the v ery interesti n g sp ecial case w ill b e treated where E / T is semi- ramiﬁed (and T / R is un ramiﬁed) and Gal( E 0 / T 0 ) is bicyclic. This case was the setting of essen tially all of Platono v’s sp eciﬁcally co m p uted exa m p les with non trivial SK 1 ( D ) [ P 2 , P 3 ], and lik ewise Y anchevski ˘ ı’s unitary examples in [ Y 3 ] wh ere the non trivial SK 1 ( D , τ ) was fully computed. This case is n ot pur sued here b ecause it r equires s ome more sp ecialized argumen ts. F or s u c h an E , it is kn o wn th at [ E ] decomp oses (non un iquely) as [ I ⊗ T N ] in the graded Brauer group of T , where I i s iner tial ov er T and N is n icely semiramiﬁed, i.e., semiramiﬁed and con taining a maximal graded subﬁ eld totally r amiﬁed ov er T . Then a form u la will b e giv en for SK 1 ( E ) as a facto r group of the relativ e Brauer group Br( E 0 / T 0 ) mod u lo other relativ e Brauer grou p s and the class of I 0 . An exactly analogous form u la will b e prov ed f or SK 1 ( E , τ ) in the unitary setting. UNIT AR Y SK 1 OF DIVISION ALGEBRAS 3 2. Preliminaries Throughout this pap er we will b e concerned with in volutory division algebras and inv olutory graded division a lgebras. In the non-graded set ting, we will denote a division algebra b y D and its cen ter b y K ; this D is equipp ed with an inv olution τ , and we set F = K τ = { a ∈ K | τ ( a ) = a } . In t h e graded setting, w e will write E for a graded d ivision algebra w ith cen ter T , and R = T τ where τ is a graded in v olution on E . (This is co n sisten t with the n otation used in [ HaW ].) Dep ending on the con text, w e will write τ ( a ) or a τ for the action of the inv olution on an elemen t, and K τ for th e set of elemen ts of K inv ariant un d er τ . Ou r con v ention is that a στ means σ ( τ ( a )). In this section, we recall the notion of graded division algebras and collect the facts w e need ab out them in § 2.1 . W e will then intro d uce the unitary and graded red u ced unitary Whitehead groups in § 2.2 and § 2.3 . 2.1. Graded division algebras. In this sub s ection we establish n otatio n an d recall some fund amen tal facts ab out graded d ivision algebras indexed by a tota lly ordered ab elian group. F or an extensiv e study of suc h graded division algebras and their relations with v alued division algebras, we refer the reader to [ HW 2 ]. F or ge n eralities on graded r ings see [ NvO ]. Let R = L γ ∈ Γ R γ b e a graded ring, i.e., Γ is an ab elian group, and R is a unital rin g such that eac h R γ is a subgroup of ( R , +) a n d R γ · R δ ⊆ R γ + δ for all γ , δ ∈ Γ. Set Γ R = { γ ∈ Γ | R γ 6 = 0 } , the grade set o f R ; R h = S γ ∈ Γ R R γ , t h e set of homogeneous element s of R . F or a homogeneous elemen t of R of degree γ , i.e., an r ∈ R γ \ { 0 } , we write deg( r ) = γ . Recall that R 0 is a subring of R and that for eac h γ ∈ Γ R , the group R γ is a left and righ t R 0 -mo dule. A sub ring S of R is a gr ade d sub ring if S = L γ ∈ Γ R ( S ∩ R γ ). F or example, the cent er of R , denoted Z ( R ), is a graded subr ing of R . If T = L γ ∈ Γ T γ is another graded ring, a gr ade d ring homomorph ism is a r ing h omomorphism f : R → T with f ( R γ ) ⊆ T γ for all γ ∈ Γ. If f is also b ijectiv e, it is called a graded ring iso morp hism; we then write R ∼ = gr T . F or a graded ring R , a gr ade d left R - mo dule M is a left R -mo dule with a grading M = L γ ∈ Γ ′ M γ , wh ere the M γ are all ab elian groups and Γ ′ is a ab elian group con taining Γ, su c h th at R γ · M δ ⊆ M γ + δ for all γ ∈ Γ R , δ ∈ Γ ′ . Then, Γ M and M h are deﬁned an alogously to Γ R and R h . W e sa y that M is a gr ade d fr e e R -mo dule if it h as a base as a f ree R -mo dule consisting of h omogeneous elemen ts. A graded ring E = L γ ∈ Γ E γ is called a gr ade d divi si on ring if Γ is a torsion-fr ee ab elian group and ev ery non-zero homogeneous elemen t of E has a m ultiplicativ e in verse in E . Note that the g r ade set Γ E is actually a g r oup. Also, E 0 is a division r ing, and E γ is a 1-dimensional left and righ t E 0 v ector space for ev ery γ ∈ Γ E . Set E ∗ γ = E γ \ { 0 } . Th e requir ement that Γ b e torsion-free is m ade b ecause w e are inte rested in graded division r ings arisin g from v aluations on division rin gs, an d all the grade groups app earing there are torsion-free. Recall that ev ery torsion-free ab elian group Γ admits total orderings compatible with the group structure. (F or example, Γ em b eds in Γ ⊗ Z Q whic h can b e given a lexicographic total ordering using an y base of it as a Q -v ector space.) By us in g an y total ord ering on Γ E , it is easy to see that E has no zero d ivisors and that E ∗ , the multiplica tive group of un its of E , coincides with E h \ { 0 } (cf. [ HW 2 , p. 78]). F urthermore, the degree map deg : E ∗ → Γ E (2.1) is a group epimorph ism w ith k ernel E ∗ 0 . By an easy adaptation of the ungraded arguments, one can see that eve r y graded mo dule M o v er a graded division ring E is graded free, and ev ery tw o homogenous bases ha v e the same cardinalit y . W e thus call M a gr ade d ve ctor sp ac e o ve r E and write dim E ( M ) f or the r ank of M as a graded f r ee E -mo du le. Let 4 R. H AZRA T AND A. R. W AD SW OR TH S ⊆ E b e a graded subring whic h is also a graded d ivision rin g. Then we can view E as a graded left S -v ector space , and we write [ E : S ] for dim S ( E ). It is easy to c heck the “F und amen tal Equality ,” [ E : S ] = [ E 0 : S 0 ] | Γ E : Γ S | , (2.2) where [ E 0 : S 0 ] is the d imension of E 0 as a left v ector space ov er the division rin g S 0 and | Γ E : Γ S | denotes the index in th e group Γ E of its subgroup Γ S . A gr ade d ﬁeld T is a comm u tativ e graded division ring. Such a T is an int egral domain (as Γ T is torsion free), so it has a q u otien t ﬁeld, which w e denote q ( T ). It is kn o wn, see [ HW 1 , Cor. 1.3], that T is in tegrally closed in q ( T ). An extensive theory of graded algebraic ﬁeld extensions of graded ﬁ elds h as b een dev elop ed in [ HW 1 ]. If E is a graded division r ing, then its cen ter Z ( E ) is clearly a graded ﬁeld. The gr ade d division rings c on- sider e d in this p ap er wil l always b e assume d ﬁnite-dimensional over their c enters. The ﬁnite-dimensionalit y assures that E has a quotien t division ring q ( E ) obtained b y central lo calization, i.e., q ( E ) = E ⊗ T q ( T ), where T = Z ( E ). Clearly , Z ( q ( E )) = q ( T ) and ind( E ) = in d( q ( E )), w here the index of E is deﬁned b y ind( E ) 2 = [ E : T ] (see [ HW 2 , p. 89]). If S is a graded ﬁeld whic h is a graded subrin g of Z ( E ) and [ E : S ] < ∞ , then E is said to b e a gr ade d division algebr a ov er S . W e recall a f undamenta l connection b et ween Γ E and Z ( E 0 ): Th e ﬁ eld Z ( E 0 ) is Galois o ver T 0 , and there is a well- d eﬁned group epimorp hism Θ E : Γ E → Gal( Z ( E 0 ) / T 0 ) giv en by deg( e ) 7→ ( z 7→ ez e − 1 ) , (2.3) for an y e ∈ E ∗ . (See [ HW 2 , Prop. 2.3] for a p ro of ). Let E = L α ∈ Γ E E α b e a graded d ivision algebra with a graded cen ter T (with, as alwa ys, Γ E a torsion-free ab elian group). After ﬁxing s ome to tal ordering on Γ E , deﬁne a fun ction λ : E \ { 0 } → E ∗ b y λ ( P c γ ) = c δ , where δ is minimal among the γ ∈ Γ E with c γ 6 = 0 . Note that λ ( a ) = a for a ∈ E ∗ , and λ ( ab ) = λ ( a ) λ ( b ) for all a, b ∈ E \ { 0 } . (2.4) Let Q = q ( E ) = E ⊗ T q ( T ), whic h is a d ivision r ing as E has no zero divisors and is ﬁ nite-dimensional o v er T . W e can extend λ to a m ap deﬁn ed on all o f Q ∗ = Q \ { 0 } as follo ws: for q ∈ Q ∗ , write q = ac − 1 with a ∈ E \ { 0 } , c ∈ Z ( E ) \ { 0 } , and set λ ( q ) = λ ( a ) λ ( c ) − 1 . It follo ws from ( 2.4 ) that λ : Q ∗ → E ∗ is w ell-deﬁned and is a group homomorphism. Since the comp osition E ∗ ֒ → Q ∗ λ − → E ∗ (2.5) is the identit y , λ is a splitting map for the in jection E ∗ ֒ → Q ∗ . F or a graded d ivision algebra E o v er its cen ter T , there is a reduced n orm map Nrd E : E ∗ → T ∗ (see [ HaW , § 3]) suc h that for a ∈ E one has Nrd E ( a ) = Nrd q ( E ) ( a ). The r e duc e d Whitehe ad gr oup , SK 1 ( E ), is deﬁn ed as E (1) / E ′ , w h ere E (1) denotes the set of elements of E ∗ with redu ced norm 1, and E ′ is the commutato r subgroup [ E ∗ , E ∗ ] of E ∗ . This group w as stud ied in detail in [ HaW ]. W e will b e using the follo wing facts whic h were established in that p ap er: R emarks 2.1 . Let n = ind( E ). (i) F or γ ∈ Γ E , if a ∈ E γ then Nrd E ( a ) ∈ E nγ . In particular, E (1) is a subset of E 0 . (ii) If S is any graded subﬁeld of E cont ainin g T and a ∈ S , then Nrd E ( a ) = N S / T ( a ) n/ [ S : T ] . (iii) Set ∂ = in d( E )  ind( E 0 ) [ Z ( E 0 ) : T 0 ]  . (2.6) If a ∈ E 0 , then, Nrd E ( a ) = N Z ( E 0 ) / T 0 Nrd E 0 ( a ) ∂ ∈ T 0 . (2.7) UNIT AR Y SK 1 OF DIVISION ALGEBRAS 5 (iv) If N is a normal subgroup of E ∗ , then N n ⊆ Nrd E ( N )[ E ∗ , N ]. F or pr o ofs of (i)-(iv) see [ HaW , P rop. 3.2 and 3.3]. (v) SK 1 ( E ) is n -torsion. Pr o of. By taking N = E (1) , the assertion foll ows from ( iv ).  A graded division algebra E with cente r T is said to b e inertial (or u nr amiﬁe d ) if Γ E = Γ T . F rom ( 2.2 ), it then follo w s that [ E : T ] = [ E 0 : T 0 ]; ind eed, E 0 is cen tral simple o ver T 0 and E ∼ = gr E 0 ⊗ T 0 T . A t the other extreme, E is said to b e tota l ly r amiﬁe d if E 0 = T 0 . In an in termediate case E is said to b e semir amiﬁe d if E 0 is a ﬁeld and [ E 0 : T 0 ] = | Γ E : Γ T | = ind( E ). Th ese d eﬁnitions are motiv ated by analogo u s deﬁn itions for v alued division algebras ([ W 2 ]). Indeed, if a tame v alued d ivision algebra is unr amiﬁed, semiramiﬁed, or totally ramiﬁed, then so is its asso ciated graded division algebra. Lik ewise, a graded ﬁ eld extension L of T is said to b e i nertial (or unr amiﬁe d ) if L ∼ = gr L 0 ⊗ T 0 T and the ﬁ eld L 0 is separable o ver T 0 . At the other extreme, L is total ly r amiﬁe d o ver T if [ L : T ] = | Γ L : Γ T | . A graded division algebra E is said to b e inertial ly spl i t if E h as a maximal graded subﬁeld L w ith L inertial o ver T . When this o ccurs, E 0 = L 0 , and ind( E ) = ind( E 0 ) [ Z ( E 0 ) : T 0 ] b y Lemma 2.2 b elow. In particular, if E is semiramiﬁed then E is in ertially split, E 0 is ab elian Galo is ov er T 0 , and the canonical map Θ E : Γ E → Gal( E 0 / T 0 ) h as k ern el Γ T (see ( 2.3 ) ab o ve and [ HW 2 , Prop. 2.3]). Lemma 2.2. L et E b e a gr ade d division algebr a with c enter T . F or the ∂ of ( 2.6 ) , ∂ 2 = | ker(Θ E ) / Γ T | . Also , ∂ = 1 iﬀ E is inertial ly split. Pr o of. S ince Θ E is surjectiv e, Γ T ⊆ k er (Θ E ), and Z ( E 0 ) is Galois o ver T 0 , we hav e ∂ 2 = ind( E ) 2   ind( E 0 ) 2 [ Z ( E 0 ) : T 0 ] 2  = [ E : T ]  [ E 0 : Z ( T 0 )] [ Z ( E 0 ) : T 0 ] | Gal( Z ( E 0 ) / T 0 ) |  = [ E 0 : T 0 ] | Γ E / Γ T |   [ E 0 : T 0 ] | im(Θ E ) |  = | k er (Θ E ) / Γ T | . No w, supp ose M is a maximal subﬁeld of E 0 with M sep arab le o ve r T 0 . Then, M ⊇ Z ( E 0 ) and [ M : Z ( E 0 )] = ind( E 0 ). Let L = M ⊗ T 0 T whic h is a graded subﬁ eld of E inertial o ver T , with L 0 = M . Then, [ L : T ] = [ L 0 : T 0 ] = [ L 0 : Z ( E 0 )] [ Z ( E 0 ) : T 0 ] = ind( E ) /∂ . Th u s, if ∂ = 1, then E is in er tially sp lit, sin ce L is a m aximal grad ed sub ﬁeld of E whic h is inertial o v er T . Con versely , supp ose E is inertially split, say I is a maximal graded su b ﬁeld of E with I inertial ov er T . So, [ I 0 : T 0 ] = [ I : T ] = ind( E ). Since I 0 Z ( E 0 ) is a subﬁeld of E 0 , w e h a v e [ I 0 : T 0 ] ≤ [ I 0 Z ( E 0 ) : T 0 ] = [ I 0 Z ( E 0 ) : Z ( E 0 )] [ Z ( E 0 ) : T 0 ] ≤ ind( E 0 ) [ Z ( E 0 ) : T 0 ] = ind( E ) /∂ = [ I 0 : T 0 ] /∂ . So, as ∂ is a p ositive integ er, ∂ = 1.  2.2. Unitary SK 1 of division algebras. W e b egin with a description of unitary K 1 and SK 1 for a division algebra with an inv olution. The analogous deﬁnitions for graded division algebras w ill b e giv en in § 2.3 . Let D b e a d ivision rin g ﬁn ite-dimensional ov er its cent er K of index n , and let τ b e an in vo lu tion on D , i.e., τ is an ant iautomorph ism of D with τ 2 = id. Let S τ ( D ) = { d ∈ D | τ ( d ) = d } ; Σ τ ( D ) = h S τ ( D ) ∩ D ∗ i . Note that Σ τ ( D ) is a normal s u bgroup of D ∗ . F or, if a ∈ S τ ( D ), a 6 = 0, and b ∈ D ∗ , then bab − 1 = [ baτ ( b )][ bτ ( b )] − 1 ∈ Σ τ ( D ), as baτ ( b ) , bτ ( b ) ∈ S τ ( D ). Let ϕ b e an isotropic m -dimensional, nond egenerate sk ew-Hermitian f orm o ver D with r esp ect to an in vol u tion τ on D . Let ρ b e the inv olution on M m ( D ) adjoint to ϕ , let U m ( D ) = { a ∈ M m ( D ) | aρ ( a ) = 1 } 6 R. H AZRA T AND A. R. W AD SW OR TH b e the unitary group associated to ϕ , and let EU m ( D ) denote the normal subgroup of U m ( D ) generated b y the u nitary transvec tions. F or m > 2, th e W all spin or norm map Θ : U m ( D ) → D ∗ / Σ τ ( D ) D ′ w as dev elop ed in [ W a ], where it was sho wn that ker(Θ) = EU m ( D ). He re, D ′ denotes the m ultiplicativ e comm utator group [ D ∗ , D ∗ ]. Com b in ing this with [ D , Cor. 1 of § 22] one obtains the comm utativ e diagram: U m ( D )  EU m ( D ) ∼ = Θ / /   D ∗  Σ τ ( D ) D ′  1 − τ   GL m ( D )  E m ( D ) det / / Nrd   D ∗  D ′ Nrd   K ∗ id / / K ∗ (2.8) where the map det is the Dieudonn´ e determinant and 1 − τ : D ∗ /  Σ τ ( D ) D ′  − → D ∗ /D ′ is d eﬁned as x Σ τ ( D ) D ′ 7→ x 1 − τ D ′ , where x 1 − τ means x τ ( x ) − 1 (see [ HM , 6.4.3]). F rom the diagram, and parallel to the “a b s olute” case, one deﬁn es the unitary Whitehe ad gr oup , K 1 ( D , τ ) = D ∗ /  Σ τ ( D ) D ′  . F or any inv olution τ on D , r ecall that Nrd D ( τ ( d )) = τ (Nrd D ( d )) , (2.9) for an y d ∈ D . F or, if p ∈ K [ x ] is the minimal p olynomial of d ov er K , then τ ( p ) is the minimal p olynomial of τ ( d ) o ver K (see also [ D , § 22, Lemma 5]). W e consider t wo cases: 2.2.1. Involutions of the ﬁrst kind. In th is case th e cent er K of D is elemen t wise in v arian t under the in vol u tion, i.e., K ⊆ S τ ( D ). T hen S τ ( D ) is a K -v ector space. The inv olutions of this k in d are furth er sub d ivided into t wo types: ortho g onal and symple ctic in v olutions (see [ KMR T , Def. 2.5]). By ([ KMR T , Prop. 2.6]), if char( K ) 6 = 2 an d τ is orthogonal then, dim K ( S τ ( D )) = n ( n + 1) / 2, while if τ is symplectic then dim K ( S τ ( D )) = n ( n − 1) / 2. Ho w eve r , if c har( K ) = 2, then dim K ( S τ ( D )) = n ( n + 1) / 2 for eac h t yp e. If d im K ( S τ ( D )) = n ( n + 1) / 2, th en for any x ∈ D ∗ , we hav e xS τ ( D ) ∩ S τ ( D ) 6 = 0 b y d imension coun t; it then follo ws that D ∗ = Σ τ ( D ), and thus K 1 ( D , τ ) = 1. Ho wev er, in the case dim K ( S τ ( D )) = n ( n − 1) / 2, Platono v sho wed th at K 1 ( D , τ ) is not in general trivial, set tling Dieudonn´ e’s conjecture in negativ e [ P 1 ]. Note that wheneve r τ is of the ﬁrst kin d we ha v e Nrd D ( τ ( d )) = Nrd D ( d ) for all d ∈ D , by ( 2.9 ). Thus, K 1 ( D , τ ) is sent to the identit y under the comp osition Nrd ◦ (1 − τ ). This explains why one do es not consider the kernel of this map, i.e., th e unitary SK 1 , for inv olutions of the ﬁrst kind . If c har( K ) 6 = 2 and τ is symplectic, th en as the m -dimensional form ϕ o ve r D is sk ew-Hermitian, its asso ciated adjoint in v olution ρ on M m ( D ) is of orthogo nal t yp e, so there is an asso ciated spin group Spin( M m ( D ) , ρ ). F or an y a ∈ S ( D ) one then h as Nrd D ( a ) ∈ K ∗ 2 ([ KMR T , Lemma 2.9]) . One deﬁnes K 1 Spin( D, τ ) = R ( D ) /  Σ τ ( D ) D ′  , where R ( D ) = { d ∈ D ∗ | Nrd D ( d ) ∈ K ∗ 2 } . This group is related to Spin( M m ( D ) , ρ ), and has b een studied in [ MY ], parallel to the w ork on absolute SK 1 groups and unitary SK 1 groups for un itary inv olutions. 2.2.2. Involutions of the se c ond kind ( unitary involutions ) . In this case K 6⊆ S τ ( D ). Then, let F = K τ (= K ∩ S τ ( D ) ), w hic h is a s u bﬁeld of K with [ K : F ] = 2. It w as already observed b y Dieudonn ´ e that U m ( D ) 6 = EU m ( D ). An imp ortan t prop erty prov ed by Platono v and Y anc hevski ˘ ı, wh ic h w e will use frequent ly , is th at D ′ ⊆ Σ τ ( D ) . (2.10) UNIT AR Y SK 1 OF DIVISION ALGEBRAS 7 (F or a pr o of, see [ KMR T , Pr op. 17.26].) Thus K 1 ( D , τ ) = D ∗ / Σ τ ( D ), which is not trivial in general. The k ernel of the map Nrd ◦ (1 − τ ) in diagram ( 2.8 ), is called the r e duc e d unitary Whitehe ad gr oup , and denoted b y S K 1 ( D , τ ). Using ( 2.9 ), it is straightfo r w ard to see th at SK 1 ( D , τ ) = Σ ′ τ ( D ) / Σ τ ( D ) , where Σ ′ τ ( D ) = { a ∈ D ∗ | Nrd D ( a ) ∈ F ∗ } . Note that w e use the notation S K 1 ( D , τ ) for the reduced unitary Whitehead group as opp osed to Draxl’s notation USK 1 ( D , τ ) in [ D , p. 172] and Y anc hevski ˘ ı’s n otation SUK 1 ( D , τ ) [ Y 2 ] and the notation USK 1 ( D ) in [ KMR T ]. Before we deﬁ n e the corresp ondin g groups in the graded setting, let us recall that all the groups ab o ve ﬁt in Tits’ f r amew ork [ T ] of the Whitehe ad gr oup W ( G, K ) = G K /G + K where G is an almost simple, s im p ly connected linear algebraic group d eﬁ ned o ve r an inﬁnite ﬁeld K , with c har( K ) 6 = 2, and G is isotropic o v er K . Here, G K is the set of K -rational p oin ts of G and G + K , is the sub grou p of G K , generated by the unip otent r adicals of the minimal K -parab olic subgroup s of G . In this setting, for G K = SL n ( D ), n > 1, w e h a v e W ( G, K ) = SK 1 ( D ); for τ an inv olution of ﬁrs t or second kind on D and F = K τ , for G F = SL n ( D , τ ) := SL n ( D ) ∩ U n ( D ) w e h a v e W ( G, F ) = SK 1 ( D , τ ); and for τ a sym plectic inv olution on D and ρ the adjoin t in vo lution of an m -dimensional isotropic sk ew-Hermitian form o ver D with m ≥ 3, for the spinor group G K = Spin( M m ( D ) , ρ ) w e ha ve W ( G, K ) is a d ouble co v er of K 1 Spin( D , τ ) (see [ MY ]). 2.3. Unitary SK 1 of graded division algebras. W e will now in tro du ce the unitary K 1 and SK 1 in the graded setting. Let E = L γ ∈ Γ E E γ b e a graded division ring (with Γ E a torsion-free ab elian group ) suc h th at E has ﬁ nite dimension n 2 o v er its cen ter T , a graded ﬁeld. L et τ b e a graded inv olution of E , i.e., τ is an an tiautomorphism of E with τ 2 = id and τ ( E γ ) = E γ for eac h γ ∈ Γ E . W e deﬁn e S τ ( E ) and Σ τ ( E ), analogously to the non-graded cases, as the set of el ements of E whic h are in v ariant under τ , and the multiplicat ive group generated by the nonzero h omogenous element s of S τ ( E ), r esp ectiv ely . W e sa y the in v olution of the ﬁrst kind if all the elements of the cente r T are inv arian t under τ ; it is of the se c ond kind (or u ni tary ) otherwise. If τ is of the ﬁrst kind then, parallel to the non-graded case, either dim T ( S τ ( E )) = n ( n + 1) / 2 or dim T ( S τ ( E )) = n ( n − 1) / 2. Ind eed, one can sho w these equalities by argu m en ts analogous to the nongraded case as in the p ro of of [ KMR T , Prop. 2.6(1)], as E is sp lit b y a graded maximal subﬁeld and the S k olem–No ether theorem is a v ailable in the graded setting ([ HW 2 , Prop. 1.6]). (These equalities can also b e obtained by passing to the qu otien t division algebra as is d one in Lemma 2.3 ( i ) b elo w.) Deﬁne the unitary Whitehe ad gr oup K 1 ( E , τ ) = E ∗ /  Σ τ ( E ) E ′  , where E ′ = [ E ∗ , E ∗ ]. If τ is of the ﬁ rst kind, char( T ) 6 = 2, and dim T ( S τ ( E )) = n ( n − 1) / 2, a pr o of similar to [ KMR T , Pr op . 2.9], sho ws that if a ∈ S τ ( E ) is homogeneous, then Nrd E ( a ) ∈ T ∗ 2 (This can also b e ve r iﬁed b y passin g to the quotien t division algebra, then usin g Lemma 2.3 ( i ) b elow and in v oking the corresp ond ing result for ungraded d ivision algebras.) F or this type of inv olution, deﬁne the spinor Whitehe ad gr oup K 1 Spin( E , τ ) = { a ∈ E ∗ | Nrd E ( a ) ∈ T ∗ 2 } /  Σ τ ( E ) E ′  . When the graded inv olution τ on E is un itary , i.e., τ | T 6 = id, let R = T τ , whic h is a graded su bﬁeld of T w ith [ T : R ] = 2. F u rthermore, T is Galois o ve r R , with Gal( T / R ) = { id , τ | T } . (See [ HW 1 ] for Galois theory for graded ﬁeld extensions.) De ﬁ n e the r e duc e d unitary Whitehe ad gr oup SK 1 ( E , τ ) = Σ ′ τ ( E ) / (Σ τ ( E ) E ′ ) = Σ ′ τ ( E ) / Σ τ ( E ) , (2.11) where Σ ′ τ ( E ) =  a ∈ E ∗ | Nrd E ( a 1 − τ ) = 1  = { a ∈ E ∗ | Nrd E ( a ) ∈ R ∗ } and Σ τ ( E ) = h a ∈ E ∗ | a 1 − τ = 1  = h S τ ( E ) ∩ E ∗ i . 8 R. H AZRA T AND A. R. W AD SW OR TH Here, a 1 − τ means aτ ( a ) − 1 . S ee Lemma 2.3 (iv) b elo w f or the second equalit y in ( 2.11 ). The group SK 1 ( E , τ ) will b e the main fo cu s o f the rest of the p ap er. W e will u s e th e follo w ing facts r ep eatedly: Lemma 2.3. (i) Any gr ade d involution on E extends uniquely to an involutio n of the same kind ( and typ e ) on Q = q ( E ) . (ii) F or any gr ade d involution τ on E , and its extension to Q = q ( E ) , we have Σ τ ( Q ) ∩ E ∗ ⊆ Σ τ ( E ) . (iii) If τ is a gr ade d involution of th e ﬁrst kind o n E with dim T ( S τ ( E )) = n ( n + 1) / 2 , th en Σ τ ( E ) = E ∗ . (iv) If τ is a unitary gr ade d involution on E , then E ′ ⊆ Σ τ ( E ) . (v) If τ is a unitary gr ade d involution on E , then SK 1 ( E , τ ) is a torsion gr oup of b ounde d exp onent dividing n = ind( E ) . Pr o of. (i) Let τ b e a g r ad ed in volutio n on E . Then q ( E ) = E ⊗ T q ( T ) = E ⊗ T ( T ⊗ T τ q ( T τ )) = E ⊗ T τ q ( T τ ). The unique extension of τ to q ( E ) is τ ⊗ id q ( T τ ) , wh ic h w e denote simply as τ . It then follo ws that S τ ( q ( E )) = S τ ( E ) ⊗ T τ q ( T τ ). Sin ce q ( T τ ) = q ( T ) τ , the assertion follo ws. (ii) Note that for the map λ in the sequen ce ( 2.5 ) we ha ve τ ( λ ( a )) = λ ( τ ( a )) for all a ∈ Q ∗ . Hence, λ (Σ τ ( Q )) ⊆ Σ τ ( E ). S ince λ | E ∗ is the identit y , we hav e Σ τ ( Q ) ∩ E ∗ ⊆ Σ τ ( E ). (iii) T h e extension of the graded in v olution τ to Q = q ( E ), also denoted τ , is of the ﬁrst kind with dim Q ( S τ ( Q )) = n ( n + 1) / 2 b y ( i ). Therefore Σ τ ( Q ) = Q ∗ (see § 2.2.1 ). Using ( ii ) no w, the assertion f ollo w s. (iv) Since τ is a un itary graded inv olution, its extension to Q = q ( E ) is a lso unitary , b y ( i ). But Q ′ ⊆ Σ τ ( Q ), as noted in ( 2.10 ) . F rom ( 2. 5 ) it f ollo w s that Q ′ ∩ E ∗ = E ′ . Hence, using ( ii ), E ′ ⊆ E ∗ ∩ Q ′ ⊆ E ∗ ∩ Σ τ ( Q ) ⊆ Σ τ ( E ). (v) Setting N = Σ ′ τ ( E ), Remark 2.1 ( iv ) ab ov e, coupled with the fact that E ′ ⊆ Σ τ ( E ) ( iv ), imp lies that SK 1 ( E , τ ) is an n -torsion group. This assertion also follo ws b y using ( ii ) whic h imp lies the natur al map SK 1 ( E , τ ) → SK 1 ( Q, τ ) is injectiv e and the fact that unitary SK 1 of a division algebra of index n is n -torsion ([ Y 2 , Cor. to 2.5]).  2.4. Generalized dihedral groups and ﬁeld extensions. T he n on trivial case of SK 1 ( E , τ ) for τ a unitary graded inv olution turn s out to b e when T = Z ( E ) is unr amiﬁed o v er R = T τ (see § 4.2 ). When that o ccurs, we will see in Lemma 4.6 ( ii ) b elo w that Z ( E 0 ) is a so-called generalized d ihedral extension ov er R 0 . W e now give the deﬁn ition and observe a few easy facts ab out generalized dihedr al groups and extensions. Deﬁnition 2.4. (i) A group G is said to b e gener alize d dihe dr al if G has a subgroup H such that [ G : H ] = 2 and e very τ ∈ G \ H satisﬁes τ 2 = id. Note that if G is generalized dihedral and H the distinguished s ubgroup, then H is ab elian and ( hτ ) 2 = id, for all τ ∈ G \ H and h ∈ H . Thus, τ 2 = id and τ hτ − 1 = h − 1 for all τ ∈ G \ H, h ∈ H . F urthermore, ev ery su bgroup of H is normal in G . Clearly ev ery dihedral group is generalized dihedral, as is ev ery elemen tary ab elian 2-group. More generally , if H is any a b elian group and χ ∈ Aut( H ) is the m ap h 7→ h − 1 , then the s emi-direct pro du ct H ⋊ i h χ i is a generalized d ihedral group, where i : h χ i → Aut( H ) is the inclusion map. It is easy to c heck that every generalized d ihedral group is isomorphic to such a semi-dir ect p ro duct. (ii) Let F ⊆ K ⊆ L b e ﬁelds with [ L : F ] < ∞ and [ K : F ] = 2. W e s a y th at L is gener alize d dihe dr al for K/F if L is Galois o v er F and eve r y elemen t of Ga l ( L/F ) \ Gal ( L/K ) has order 2, i.e., Gal( L/F ) is a generalized dihedr al group. Note that w hen this o ccurs, L is comp ositum of ﬁ elds L i con taining K with eac h L i generalized dihedral f or K /F with Gal( L i /K ) cyc lic, i.e., L i is Galois o ver F w ith UNIT AR Y SK 1 OF DIVISION ALGEBRAS 9 Gal( L i /F ) dih edral (or a Klein 4-group). Con ve r sely , if L and M are generalized dihedral for K /F then so is their comp ositum. Example 2.5 . Let n ∈ N , n ≥ 3, and let F ⊆ K b e ﬁelds w ith [ K : F ] = 2 and K = F ( ω ), where ω is a pr imitiv e n -th ro ot of u nit y (so c har( F ) ∤ n ). Su pp ose the n on-iden tit y elemen t of Gal( K/F ) m aps ω to ω − 1 . F or an y c 1 , . . . , c k ∈ F ∗ , if ω 6∈ F ( n √ c 1 , . . . , n √ c k ), then K ( n √ c 1 , . . . , n √ c k ) is generalized dih edral for K/F . 3. Hensel ian to graded reduction The main goal of this section is to pr ov e an isomorp hism b et wee n the unitary S K 1 of a v alued division algebra w ith in volution o ver a henselian ﬁeld and the graded SK 1 of its asso ciated graded d ivision algebra. W e ﬁ rst r ecall ho w to asso ciate a graded division algebra to a v alued division algebra. Let D b e a d ivision alg ebr a ﬁn ite dimensional ov er its cen ter K , with a v aluation v : D ∗ → Γ. So, Γ is a totally ordered ab elian group, and v satisiﬁes the conditions th at for all a, b ∈ D ∗ , (1) v ( ab ) = v ( a ) + v ( b ); (2) v ( a + b ) ≥ min { v ( a ) , v ( b ) } ( b 6 = − a ) . Let V D = { a ∈ D ∗ | v ( a ) ≥ 0 } ∪ { 0 } , the v aluation rin g of v ; M D = { a ∈ D ∗ | v ( a ) > 0 } ∪ { 0 } , the un ique maximal left (and right) ideal of V D ; D = V D / M D , the resid u e division ring of v on D ; and Γ D = im( v ) , th e v alue group of the v aluation . No w let K b e a ﬁeld with a v aluation v , and supp ose v is henselian ; that is, v has a unique extension to ev ery algebraic ﬁeld extension of K . Recall that a ﬁ eld extension L of K of degree n < ∞ is said to b e ta mely r amiﬁe d or tame o ver K if, with resp ect to the unique extension of v to L , the residue ﬁeld L is separable o v er K and c har( K ) ∤  n  [ L : K ]  . Su c h an L is necessarily defe ctless o ver K , i.e., [ L : K ] = [ L : K ] | Γ L : Γ K | , by [ EP , Th. 3.3.3] (applied to N /K and N/L , where N is a normal closure of L ov er K ). Along the same lines, let D b e a division algebra with cente r K (so, b y con v ention, [ D : K ] < ∞ ); then the henselian v aluation v on K extends uniqu ely to a v aluation on D ([ W 1 ]). With resp ect to this v aluation, D is said to b e tamely r amiﬁe d or tame if the cent er Z ( D ) is separable o ver K and c har( K ) ∤  ind( D )  ind( D )[ Z ( D ) : K ]  . Recall from [ JW , Prop. 1.7 ], that whenever the ﬁeld extension Z ( D ) /K is separable, it is ab elian Galois. It is kno wn that D is tame if and only if D is split b y the maximal tamely ramiﬁed ﬁ eld extension of K , if and only if c har( K ) = 0 or char( K ) = p 6 = 0 and the p -primary comp onent of D is inertially sp lit, i.e., split by the maximal unramiﬁed extension of K ([ JW , Lemma 6.1]). W e sa y D is str ongly tame if c har( K ) ∤ ind( D ). Note that strong tameness implies tameness. This is clear from the last characte r ization of tameness, or f rom ( 3.1 ) b elo w . Recall also fr om [ Mor , Th. 3], that for a v alued division algebra D ﬁnite dimensional o ver its cen ter K (here not necessarily henselian), w e h a v e the “Ostr o wski theorem” [ D : K ] = q k [ D : K ] | Γ D : Γ K | , (3.1) where q = c har ( D ) and k ∈ Z with k ≥ 0 (and q k = 1 if char( D ) = 0). I f q k = 1 in e qu ation ( 3.1 ), then D is said to b e defe ctless o ve r K . F or bac kground on v alued d ivision alge b ras, see [ JW ] or the sur v ey pap er [ W 2 ]. R emark 3.1 . If a ﬁ eld K has a h enselian v aluation v and L is a sub ﬁeld of K with [ K : L ] < ∞ , then the restriction w = v | L need not b e henselian. But it is easy to see that w is then “semihenselian,” i.e., w h as more than one but only ﬁnitely many diﬀerent extensions to a separable closure L sep of L . See [ En ] 10 R. H AZRA T AND A. R. W AD SW OR TH for a thorough analysis of semihenselian v aluations. Notably , Engler sho ws that w is semihenselian iﬀ the residue ﬁeld L w is algebraical ly closed but there is a h enselian v aluation u on L such that u is a prop er coarsening of w a n d the residue ﬁ eld L u is real cl osed. When this o ccurs, char( L ) = 0, L is formally real , w h as exactly t wo extensions to L sep , the v alue grou p Γ L,w has a nontrivia l divisible su bgroup, and the henselization of L re w is L ( √ − 1), whic h lies in K . F or example, if we tak e an y prime num b er p , let w p b e the p -adic discrete v aluation on Q , and let L = { r ∈ R | r is algebraic o ver Q } ; then any extension of w p to L is a semihen selian v aluation. Note that if v on K is discrete, i.e., Γ K ∼ = Z , then w on L cannot b e semihenselian, since Γ L has no n on trivial divisible sub group; so, w on L m u st b e henselian. T his preserv ation of the henselian prop ert y for discrete v aluations was asserted in [ Y 2 , Lemma, p. 195], but th e pro of giv en there is in v alid. One asso ciates to a v alued division algebra D a graded d ivision algebra as follo ws: F or eac h γ ∈ Γ D , let D ≥ γ = { d ∈ D ∗ | v ( d ) ≥ γ } ∪ { 0 } , an additive subgroup of D ; D >γ = { d ∈ D ∗ | v ( d ) > γ } ∪ { 0 } , a su bgroup of D ≥ γ ; and gr ( D ) γ = D ≥ γ  D >γ . Then deﬁne gr ( D ) = L γ ∈ Γ D gr ( D ) γ . Because D >γ D ≥ δ + D ≥ γ D >δ ⊆ D > ( γ + δ ) for all γ , δ ∈ Γ D , the m ultiplication on gr ( D ) induced b y m ulti- plication on D is we ll-deﬁn ed, giving that gr ( D ) is a graded ring, calle d the asso ciate d gr ade d ring of D . The multiplica tive p r op ert y (1) of the v aluation v implies th at gr ( D ) is a graded division ring. Clearly , we ha ve gr ( D ) 0 = D and Γ gr ( D ) = Γ D . F or d ∈ D ∗ , we write e d for the image d + D >v ( d ) of d in gr ( D ) v ( d ) . Th u s, the map giv en b y d 7→ e d is a group epimorph ism ρ : D ∗ → gr ( D ) ∗ with ke r nel 1 + M D , giving us the short exact sequence 1 − → 1 + M D − → D ∗ − → gr ( D ) ∗ − → 1 , (3.2) whic h will b e us ed throughout. F or a detailed study of the a ss o ciated graded algebra of a v alued d ivision algebra r efer to [ HW 2 , § 4]. As sho w n in [ HaW , C or. 4.4], the r educed norm m aps f or D and gr ( D ) are related b y ^ Nrd D ( a ) = Nrd gr ( D ) ( e a ) for all a ∈ D ∗ . (3.3) No w let K b e a ﬁ eld with a henselian v aluation v and , as b efore, let D b e a division algebra w ith cen ter K . Then v extends un iquely to a v aluation on D , also d enoted v , and one obtains asso ciated to D the graded d ivision algebra gr ( D ) = L γ ∈ Γ D D γ . F urther, su pp ose D is tame with resp ect to v . This imp lies that [ gr ( D ) : gr ( K )] = [ D : K ], gr ( K ) = Z ( gr ( D )) and D has a maximal subﬁeld L with L tamely ramiﬁed o v er K ([ HW 2 , Prop. 4.3]). W e can then asso ciate to an inv olution τ on D , a graded in vol u tion e τ on gr ( D ). First, supp ose τ is of th e ﬁr st kind on D . Then v ◦ τ is also a v aluation on D whic h restricts to v on K ; then, v ◦ τ = v since v has a uniqu e extension to D . S o, τ induces a we ll-deﬁn ed map e τ : gr ( D ) → gr ( D ), deﬁn ed on homogeneous elemen ts b y e τ ( e a ) = g τ ( a ) for all a ∈ D ∗ . Clearly , e τ is a w ell-deﬁned graded in v olution o n gr ( D ); it is of the ﬁrst kind, as it lea ves Z ( gr ( D )) = gr ( K ) in v ariant. If τ is a unitary inv olution on D , let F = K τ . In this case, we need to assum e that the restriction of the v aluation v from K to F induces a henselian v aluation on F , and th at K is tamely ramiﬁed o v er F . Since ( v ◦ τ ) | F = v | F , an argument similar to the on e ab o ve sho w s that v ◦ τ co in cides with v on K and thus on D , and the indu ced map e τ on gr ( D ) as ab o v e is a graded in volutio n . Th at K is tamely r amiﬁed o v er F means t h at [ K : F ] = [ gr ( K ) : gr ( F )], K is separable o v er F , and c har( F ) ∤ | Γ K : Γ F | . Since [ K : F ] = 2, K is alw a ys tamely ramiﬁed ov er F if c har( F ) 6 = 2. But if char( F ) = 2, K is tamely ramiﬁed o v er F if and only if [ K : F ] = 2, Γ K = Γ F , and K is separable (so Ga lois) ov er F . Since K is Galo is ov er F , the canonical map Gal( K/F ) → Gal( K /F ) is su rjectiv e, b y [ EP , pp. 123–124, pro of of Lemma 5.2.6(1)]. Hence, UNIT AR Y SK 1 OF DIVISION ALGEBRAS 11 τ indu ces the nonid en tit y F -automorphism τ of K . Also e τ is u nitary , i.e., e τ | gr ( K ) 6 = id. This is ob vious if c har( F ) 6 = 2, since then K = F ( √ c ) for some c ∈ F ∗ , and e τ ( f √ c ) = ^ τ ( √ c ) = − f √ c 6 = f √ c . If c har( F ) = 2, then K is u nramiﬁed o ver F and e τ | gr ( K ) 0 = τ (the automorphism of K induced b y τ | K ) wh ic h is non trivial as Gal( K/F ) maps onto Gal( K /F ); so again e τ | gr ( K ) 6 = id. Th u s, e τ is a unitary graded inv olution in an y characte r istic. Moreo v er, for the graded ﬁ xed ﬁeld gr ( K ) e τ w e ha ve gr ( F ) ⊆ gr ( K ) e τ $ gr ( K ) and [ gr ( K ) : gr ( F )] = 2, so gr ( K ) e τ = gr ( F ). Theorem 3.2. L et ( D , v ) b e a tame value d division algebr a over a henselian ﬁeld K , with c har( K ) 6 = 2 . If τ is an involution of the ﬁrst kind o n D , then K 1 ( D , τ ) ∼ = K 1 ( gr ( D ) , e τ ) , and if τ is symple c tic, then K 1 Spin( D , τ ) ∼ = K 1 Spin( gr ( D ) , e τ ) . Pr o of. Let ρ : D ∗ → gr ( D ) ∗ b e the group epimorph ism giv en in ( 3.2 ). Clearly ρ ( S τ ( D )) ⊆ S e τ ( gr ( D )), so ρ (Σ τ ( D )) ⊆ Σ e τ ( gr ( D )). Consid er the foll owing diagram: 1 / / (1 + M D ) ∩ Σ τ ( D ) D ′ / /   Σ τ ( D ) D ′ ρ / /   Σ e τ ( gr ( D )) gr ( D ) ′ / /   1 1 / / (1 + M D ) / / D ∗ ρ / / gr ( D ) ∗ / / 1 . (3.4) The top row of the diagram is exact . T o see this, n ote that ρ ( D ′ ) = gr ( D ) ′ . Thus, it su ﬃces to sho w that ρ maps S τ ( D ) ∩ D ∗ on to S e τ ( gr ( D )) ∩ gr ( D ) ∗ . F or this, tak e an y d ∈ D ∗ with e d = e τ ( e d ). L et b = 1 2 ( d + τ ( d )) ∈ S τ ( D ). Since v ( b ) = v ( τ ( b )) and e d + g τ ( d ) = 2 e d 6 = 0, e b = 1 2 ( ^ d + τ ( d )) = 1 2 ( e d + g τ ( d )) = e d . Since τ on D is an inv olution of the ﬁrs t kind, the index of D is a p o w er of 2 ([ D , T h. 1, § 16]). As c har( K ) 6 = 2, it follo ws that the v aluation is strongly tame, and b y [ Ha , Lemma 2.1], 1 + M D = (1 + M K )[ D ∗ , 1 + M D ] ⊆ Σ τ ( D ) D ′ . Therefore, the left v ertical map is the iden tit y map. It follo w s (for example using t h e sn ak e lemma) that K 1 ( D , τ ) ∼ = K 1 ( gr ( D ) , e τ ). The pr o of for K 1 Spin when τ is of symp lectic type is similar.  The k ey to pr o ving the corresp onding resu lt f or unitary in volutio n s is the Con gru ence Theorem: Theorem 3.3 (Congruence Theorem) . L et D b e a tame division algebr a over a ﬁeld K with henselian valuation v . L et D (1) = { a ∈ D ∗ | Nrd D ( a ) = 1 } . Then, D (1) ∩ (1 + M D ) ⊆ [ D ∗ , D ∗ ] . This theorem w as pro ved b y Platono v in [ P 2 ] for v a complete discrete v aluation, and it was an essen tial to ol in al l his c alculations of SK 1 for division rings. Th e Congruence Theorem w as asserted by Er sho v in [ E ] in the generalit y giv en h ere. A full pr o of is giv en in [ HaW , T h. B.1]. Prop osition 3.4 (Unitary Congruence Theorem) . L e t D b e a tame division algebr a over a ﬁeld K with henselian va luation v , and let τ b e a unitary involution on D . L et F = K τ . If F is henselian with r e sp e ct to v | F and K is ta mely r amiﬁe d over F , then (1 + M D ) ∩ Σ ′ τ ( D ) ⊆ Σ τ ( D ) . 12 R. H AZRA T AND A. R. W AD SW OR TH Pr o of. Th e only pub lished pr o of of this w e kno w is [ Y 2 , Th. 4.9], whic h is just for the case v discrete rank 1; that proof is r ather h ard to follo w, and app ears to apply for other v aluations only if D is inertially split. Here w e pro vide another pr o of, in fu ll generalit y . W e use the well-kno w n facts th at Nrd D (1 + M D ) = 1 + M K and N K/F (1 + M K ) = 1 + M F . (3.5) (The second equ ation holds as K is tamely ramiﬁed ov er F .) See [ E , Prop. 2] or [ HaW , Prop. 4.6, Cor. 4.7] for a pro of. No w, ta ke m ∈ M D with Nrd D (1 + m ) ∈ F . T hen Nrd D (1 + m ) ∈ F ∩ (1 + M K ) = 1 + M F . By ( 3.5 ) there is c ∈ 1 + M K with Nrd D (1 + m ) = N K/F ( c ) = cτ ( c ), and there is b ∈ 1 + M D with Nrd D ( b ) = c . Then, Nrd D ( bτ ( b )) = cτ ( c ) = N K/F ( c ) = Nrd D (1 + m ) . Let s = (1 + m )( bτ ( b )) − 1 ∈ 1 + M D . Since Nrd D ( s ) = 1, by the Congruence Th eorem for SK 1 , T h . 3.3 ab o ve, s ∈ [ D ∗ , D ∗ ] ⊆ Σ τ ( D ), (recall ( 2.10 )) . Since bτ ( b ) ∈ S τ ( D ), we ha v e 1 + m = s ( bτ ( b )) ∈ Σ τ ( D ).  Theorem 3.5. L et D b e a tame division algebr a over a ﬁeld K with henselian v aluation v . L et τ b e a unitary involution on D , and let F = K τ . If F is henselian with r esp e ct to v | F and K is tamely r amiﬁe d over F , then τ induc es a unitar y gr ade d involution e τ of gr ( D ) with gr ( F ) = gr ( K ) e τ , and SK 1 ( D , τ ) ∼ = SK 1 ( gr ( D ) , e τ ) . Pr o of. Th at e τ is a unitary graded inv olution on gr ( D ) an d gr ( F ) = gr ( K ) e τ w as already observ ed (see the discussion b efore Th. 3.2 ). F or the canonical epimorph ism ρ : D ∗ → gr ( D ) ∗ , a 7→ e a , it follo ws f r om ( 3.3 ) that ρ (Σ ′ τ ( D ) ⊆ Σ ′ e τ ( gr ( D )). Also, clearly ρ ( S τ ( D )) ⊆ S e τ ( gr ( D )), so ρ (Σ τ ( D )) ⊆ Σ e τ ( gr ( D )). Thus, there is a comm utativ e diagram 1 / / (1 + M D ) ∩ Σ τ ( D )   / / Σ τ ( D )   ρ / / Σ e τ ( gr ( D ))   / / 1 1 / / (1 + M D ) ∩ Σ ′ τ ( D ) / / Σ ′ τ ( D ) ρ / / Σ ′ e τ ( gr ( D )) / / 1 , (3.6) where the v ertical maps are inclusions, and the left vertica l map is bijectiv e, b y Pr op . 3.4 ab o ve . T o see that the b ottom ro w of diagram ( 3.6 ) is exact at Σ ′ e τ ( gr ( D )), tak e b ∈ D with Nrd gr ( D ) ( e b ) ∈ gr ( F ). Let c = Nr d D ( b ) ∈ K ∗ . Th en e c = Nrd gr ( D ) ( e b ) ∈ gr ( F ), so e c = e t for some t ∈ F ∗ . Let u = c − 1 t ∈ 1 + M K . By ( 3.5 ) abov e, there is d ∈ 1 + M D with Nrd D ( d ) = u . So, Nrd D ( bd ) = cu = t ∈ F ∗ . Th u s , bd ∈ Σ ′ τ ( D ) and ρ ( bd ) = f bd = e b . This giv es the claimed exactness, and sho w s that the b ottom ro w of d iagram ( 3.6 ) is exact. T o see that the top ro w of diagram ( 3.6 ) is exact at Σ e τ ( gr ( D )), it suﬃ ces to show that ρ m ap s S τ ( D ) ∩ D ∗ on to S e τ ( gr ( D )) ∩ gr ( D ) ∗ . F or this, ta ke an y d ∈ D ∗ with e d = e τ ( e d ). If char( F ) 6 = 2, as in the pr o of of Th. 3.2 , let b = 1 2 ( d + τ ( d )) ∈ S τ ( D ). Since v ( b ) = v ( τ ( b )) and e d + g τ ( d ) = 2 e d 6 = 0, we ha ve e b = 1 2 ( ^ d + τ ( d )) = 1 2 ( e d + g τ ( d )) = e d . If c har ( F ) = 2, then K is unramiﬁed o v er F , so K is Galois o ver F with [ K : F ] = 2, and the map τ : K → K induced by τ is the n oniden tit y F -automorphism of K . Of course, K = gr ( K ) 0 and τ = e τ | gr ( K ) 0 . Because K is separable o v er F , the trace tr K /F is surj ectiv e, so there is r ∈ V K with e r + e τ ( e r ) = 1 ∈ gr ( F ) 0 . Let c = r d + τ ( r d ) ∈ S τ ( D ). W e ha ve f r d = e r e d and ] τ ( r d ) = e τ ( f r d ) = e τ ( e r e d ) = e τ ( e d ) e τ ( e r ) = e τ ( e r ) e d. Since v ( r d ) = v ( τ ( r d )) and f r d + ] τ ( r d ) = e r e d + e τ ( e r ) e d = e d 6 = 0, we h av e e c = f r d + ] τ ( r d ) = e d . So, in all cases ρ ( S τ ( D ) ∩ D ∗ ) = S e τ ( gr ( D )) ∩ gr ( D ) ∗ , fr om whic h it f ollo w s that the b ottom row of diagram ( 3.6 ) is exact. UNIT AR Y SK 1 OF DIVISION ALGEBRAS 13 Since eac h ro w of ( 3.6 ) is exact, we ha v e a righ t exact sequence of cok ernels of the vertic al m aps, which yields the isomorphism o f th e theorem.  Ha ving established the bridge b etw een the unitary K -groups in the graded setting and the non-graded henselian case (Th. 3.2 , Th. 3.5 ), we can deduce known formula s in the literature for the unitary Whitehead group of certain v alued division algebras, by passing to the graded setting. Th e p r o ofs are muc h easier than those previously a v ailable. W e will do this systematically for unitary in volutio n s in Section 4 . Before w e tu rn to that, here is an example with an in v olution of the ﬁ rst kin d: Example 3.6 . Let E b e a graded division algebra o v er its cen ter T with an in vol u tion τ of the ﬁ rst kind . If E is unr amiﬁed ov er T , then, b y u sing E ∗ = E ∗ 0 T ∗ , it follo ws easil y that K 1 ( E , τ ) ∼ = K 1 ( E 0 , τ | E 0 ) , (3.7) and, if c har ( E ) 6 = 2 and τ is symplectic, K 1 Spin( E , τ ) ∼ = K 1 Spin( E 0 , τ | E 0 ) . (3. 8) No w if D is a tame and unramiﬁ ed division algebra o ve r a h enselian v alued ﬁeld and D has an in vol u tion τ of the ﬁrst kind , then the asso ciated g r ad ed division ring gr ( D ) is also u nramiﬁed with the corresp onding graded in volutio n e τ of the ﬁ r st kind; then Th. 3.2 and ( 3.7 ) ab ov e sho w that K 1 ( D , τ ) ∼ = K 1 ( gr ( D ) , e τ ) ∼ = K 1 ( gr ( D ) 0 , τ | gr ( D ) 0 ) = K 1 ( D , τ ) , yielding a theo r em of Plat onov-Y anc h evski ˘ ı [ PY , Th. 5 .11] (that K 1 ( D , τ ) ∼ = K 1 ( D , τ ) w hen D is un ram- iﬁed o ve r K and the v aluation is henselian and discrete r ank 1.) Similarly , when c har( D ) 6 = 2 and τ is symplectic, K 1 Spin( D, τ ) ∼ = K 1 Spin( gr ( D ) , e τ ) ∼ = K 1 Spin( gr ( D ) 0 , τ | gr ( D ) 0 ) = K 1 Spin( D , τ ) . R emark 3.7 . W e ha ve the follo w in g comm utativ e diagram connecting unitary SK 1 to non-unitary S K 1 , where SH 0 ( D , τ ) and SH 0 ( D ) are the cok ern els of Nrd ◦ (1 − τ ) and Nrd r esp ectiv ely ( s ee diag r am ( 2.8 )). 1 / / SK 1 ( D , τ ) / /   D ∗ / Σ( D ) Nrd ◦ (1 − τ ) / / 1 − τ   K ∗ / / id   SH 0 ( D , τ ) / /   1 1 / / SK 1 ( D ) / / D ∗ /D ′ Nrd / / K ∗ / / SH 0 ( D ) / / 1 . (3.9) No w, let D b e a tame v alued division algebra with cen ter K and with a unitary inv olution τ , suc h that the v aluation restricts to a henselian v aluation on F = K τ . By Th. 3.5 , SK 1 ( D , τ ) ∼ = SK 1 ( gr ( D ) , e τ ) and b y [ HaW , Th. 4.8, Th. 4.12], SK 1 ( D ) ∼ = SK 1 ( gr ( D )) and SH 0 ( D ) ∼ = SH 0 ( gr ( D )). Ho wev er, SH 0 ( D , τ ) is not stable un der “v alued ﬁ ltration”, i.e., SH 0 ( D , τ ) 6 ∼ = SH 0 ( gr ( D ) , e τ ). In fact using ( 3.2 ), w e can build a comm utativ e diagram with exact rows, 1 / / (1 + M K ) ∩ Nrd ( D ∗ ) 1 − τ / /  _   Nrd ( D ∗ ) 1 − τ / /  _   Nrd  gr ( D ) ∗  1 − e τ / /  _   1 1 / / 1 + M K / / K ∗ / / gr ( K ) ∗ / / 1 , whic h in duces the exact sequence 1 − → (1 + M K )  (1 + M K ) ∩ Nrd ( D ∗ ) 1 − τ  − → SH 0 ( D , τ ) − → SH 0 ( gr ( D ) , e τ ) − → 1 . By considering the norm N K/F : K ∗ → F ∗ , we clearly h a v e Nrd ( D ∗ ) 1 − τ ⊆ ker N K/F . Ho w eve r, b y ( 3.5 ), N K/F : 1 + M K → 1 + M F is surjective , whic h s h o ws that 1 + M K  (1 + M K ) ∩ Nrd ( D ∗ ) 1 − τ  is n ot tr ivial and th u s SH 0 ( D , τ ) 6 ∼ = SH 0 ( gr ( D ) , e τ ) . 14 R. H AZRA T AND A. R. W AD SW OR TH 4. Graded Unit ar y SK 1 Calculus Let E b e a graded division algebra o ver its cente r T with a un itary graded inv olution τ , and let R = T τ . Since [ T : R ] = 2 = [ T 0 : R 0 ] | Γ T : Γ R | , there are jus t t w o possible cases: • T is totally ramiﬁ ed o v er R , i.e., | Γ T : Γ R | = 2 • T is unramﬁ ed ov er R , i.e., | Γ T : Γ R | = 1. W e will consider S K 1 ( E , τ ) in these t w o cases s eparately in § 4.1 and § 4.2 . The follo w ing notation will b e used throughout this section and the next: Let τ ′ b e another in vo lu tion on E . W e w r ite τ ′ ∼ τ if τ ′ | Z ( E ) = τ | Z ( E ) . F or t ∈ E ∗ , let ϕ t denote the m ap from E to E giv en by conjugation b y t , i.e., ϕ t ( x ) = txt − 1 . Let Σ 0 = Σ τ ∩ E ∗ 0 and Σ ′ 0 = Σ ′ τ ∩ E ∗ 0 . W e ﬁrst collect some facts which will b e used b elo w . Th ey all foll ow b y ea sy cal culations. R emarks 4.1 . (i) W e ha ve τ ′ ∼ τ if a n d o n ly if there is a t ∈ E ∗ with τ ( t ) = t and τ ′ = τ ϕ t . (Th e proof is analo gous to the ungraded version giv en, e.g. in [ KMR T , Prop. 2.18]. ) (ii) If τ ′ ∼ τ , then Σ τ ′ = Σ τ and Σ ′ τ ′ = Σ ′ τ ; th us SK 1 ( E , τ ′ ) = S K 1 ( E , τ ). (See [ Y 1 , Lemma 1] for the analogous ungraded result.) (iii) F or an y s ∈ E ∗ , we ha ve τ ϕ s = ϕ τ ( s ) − 1 τ . Hence, τ ϕ s is an in v olution (necessarily ∼ τ ) if and only if τ ϕ s = ϕ s − 1 τ if an d only if τ ( s ) /s ∈ T . (iv) If s ∈ E ∗ γ and τ ( s ) = s , then Σ ′ τ ∩ E γ = s Σ ′ 0 and S τ ∩ E γ = s ( S τ s ∩ E 0 ) where τ s = τ ϕ s . 4.1. T / R totally ramiﬁed. Let E b e a grad ed division algebra with a unitary g r aded in v olution τ suc h that T = Z ( E ) is totally ramiﬁed o ver R = T τ . In this section we w ill sh o w th at S K 1 ( E , τ ) = 1. Note that the assump tion that T / R is totally ramiﬁed imp lies that c har( T ) 6 = 2. F or, if c h ar( T ) = 2 and T is tot ally ramiﬁed o ve r a graded subﬁ eld R with [ T : R ] = 2, then for any x ∈ T ∗ \ R ∗ , w e ha ve d eg ( x 2 ) ∈ Γ R , so x 2 ∈ R ; th us, T is pu rely inseparable o v er R . That cannot happ en here, as τ | T is a non trivial R -automorphism of T . Lemma 4.2. If T is total ly r amiﬁe d over R , then τ ∼ τ ′ for some gr ade d involution τ ′ , wher e τ ′ | E 0 is of the ﬁrst kind. Pr o of. Let Z 0 = Z ( E 0 ). Since T is to tally ramiﬁed o ve r R , T 0 = R 0 , so τ | Z 0 ∈ Gal( Z 0 / T 0 ). Since the map Θ E : Γ E → Gal( Z 0 / T 0 ) is su rjectiv e (see ( 2.3 )), there is γ ∈ Γ E with Θ E ( γ ) = τ | Z 0 . Cho ose y ∈ E ∗ γ with τ ( y ) = ± y . Then set τ ′ = τ ϕ y − 1 .  Example 4.3 . Here is a construction of exa mp les of graded d ivision algebras E with un itary graded in vo- lution τ with E totally ramiﬁ ed ov er Z ( E ) τ . W e will see b elo w th at these are all suc h examples. Let R b e an y graded ﬁeld with c har( R ) 6 = 2, and let A b e a graded division algebra with cen ter R , such that A is totally ramiﬁed ov er R with exp(Γ A / Γ R ) = 2. Let T b e a graded ﬁeld extension of R w ith [ T : R ] = 2, T totally ramiﬁed o v er R , and Γ T ∩ Γ A = Γ R . Let E = A ⊗ R T , wh ic h is a graded cen tral simp le al- gebra o v er T , as A is graded central sim p le o v er R , b y [ HW 2 , Pr op. 1.1]. But b ecause Γ T ∩ Γ A = Γ R , w e h a v e E 0 = A 0 ⊗ R 0 T 0 = R 0 ⊗ R 0 R 0 = R 0 . Since E 0 is a division ring, E m u s t b e a graded division ring, whic h is totally ramiﬁed ov er R , as E 0 = R 0 . No w , b ecause A is totally ramiﬁed ov er R , w e ha ve exp( A ) = exp(Γ A / Γ R ) = 2, and A = Q 1 ⊗ R . . . ⊗ R Q m , where eac h Q i is a graded sym b ol algebra of degree at most 2, i.e., a graded quaternion algebra. Let σ i b e a graded inv olution of the ﬁrst kin d on Q i (e.g., the canonical symplectic graded inv olution), and let ρ b e the noniden tit y R -automorphism of T . T hen, σ = σ 1 ⊗ . . . ⊗ σ m is a graded inv olution of the ﬁ rst kind on A , so σ ⊗ ρ is a unitary graded in vol u tion on E , with T τ = R . UNIT AR Y SK 1 OF DIVISION ALGEBRAS 15 Prop osition 4.4. If E is total ly r amiﬁe d over R , and E 6 = T , then Σ τ = E ∗ , so SK 1 ( E , τ ) = 1 . F u rthermor e, E and τ ar e as describ e d in Ex. 4.3 . Pr o of. W e hav e E 0 = T 0 = R 0 . F or an y γ ∈ Γ E , there is a nonzero a ∈ E γ with τ ( a ) = ǫa where ǫ = ± 1. Then, f or any b ∈ E γ , b = r a for s ome r ∈ E 0 = R 0 . Sin ce r is cen tral and symmetric, τ ( b ) = ǫb . Thus, ev ery elemen t of E ∗ is sym metric or ske w -sym metric. Ind eed, ﬁx an y t ∈ T ∗ \ R ∗ . Then τ ( t ) 6 = t , as t / ∈ R ∗ . Hence, τ ( t ) = − t . Since t is central and skew-symmetric, ev ery a ∈ E ∗ is symmetric iﬀ ta is sk ew- symmetric. Thus, E ∗ = S ∗ τ ∪ tS ∗ τ . T o see that Σ τ = E ∗ , it suﬃces to show that t ∈ Σ τ . T o see this, ta ke an y c, d ∈ E ∗ with dc 6 = cd . (They exist, as E 6 = T .) By replacing c (resp. d ) if necessary b y tc (resp. td ), w e ma y assume that τ ( c ) = c and τ ( d ) = d . Th en , dc = τ ( cd ) = ǫcd , where ǫ = ± 1; since dc 6 = cd , ǫ = − 1; hence τ ( tcd ) = tcd . Thus, t = ( tcd ) c − 1 d − 1 ∈ Σ τ ( E ), completing the p ro of th at Σ τ ( E ) = E ∗ . F or γ ∈ Γ E , let γ = γ + Γ T ∈ Γ E / Γ T . T o see the stru cture of E , recall that as E is totally ramiﬁed ov er T there is a well- d eﬁned nondegenerate Z -bilinear symp lectic pairing β : (Γ E / Γ T ) × Γ E / Γ T ) → E ∗ 0 giv en by β ( γ , δ ) = y γ y δ y − 1 γ y − 1 δ for any nonzero y γ ∈ E γ , y δ ∈ E δ . The computation ab o v e f or c and d sh o ws that im( β ) = {± 1 } . Since the pairing β is n ondegenerate by [ HW 2 , Prop. 2.1] there is a symp lectic base of Γ E / Γ T , i.e., a sub s et { γ 1 , δ 1 , . . . , γ m , δ m } of Γ E / Γ T suc h that β ( γ i , δ i ) = − 1 while β ( γ i , γ j ) = β ( δ i , δ j ) = 1 for all i, j , and β ( γ i , δ j ) = 1 w henev er i 6 = j , and Γ E = h γ 1 , δ 1 , . . . , γ m , δ m i + Γ T . Cho ose any n on zero i i ∈ E γ i and j i ∈ E δ i . The prop erties of the γ i , δ i under β tr anslate to: i i j i = − j i i i while i i i j = i j i i and j i j j = j j j i for all i, j , and i i j j = j j i i whenev er i 6 = j . Since β (2 γ i , η ) = 1 for all i and all η ∈ Γ E , eac h i 2 i is cen tral in E . But also τ ( i 2 i ) = i 2 i , as τ ( i i ) = ± i i . So, eac h i 2 i ∈ R ∗ , and lik ewise eac h j 2 i ∈ R ∗ . Let Q i = R -span { 1 , i i , j i , i i j i } in E . The r elations on the i i , j i sho w that eac h Q i is a graded quaternion algebra o v er R , and the distinct Q i cen tralize eac h other in E . Since eac h Q i is graded cen tral simple ov er R , Q 1 ⊗ R . . . ⊗ R Q m is graded cen tral simple o v er R b y [ HW 2 , Prop. 1.1]. Let A = Q 1 . . . Q m ⊆ E . The graded R -algebra epimorphism Q 1 ⊗ R . . . ⊗ R Q m → A m u st b e an isomorphism, as the domain is graded simple. If Γ T ⊆ Γ A , then T ⊆ A , since E is totally ramiﬁed ov er R . But this cannot occur, as T cen tralizes A but T % R = Z ( A ). Hence, as | Γ T : Γ R | = 2, w e must ha ve Γ T ∩ Γ A = Γ R . The graded R -algebra homomorphism A ⊗ R T → E is injectiv e since its d omain is graded simple, b y [ HW 2 , Prop. 1 .1]; it is also surjectiv e, since E 0 = R 0 ⊆ A ⊗ R T and Γ A ⊗ R T ⊇ h γ 1 , δ 1 , . . . , γ m , δ m i + Γ T = Γ E . Clearly , τ = τ | A ⊗ τ | T .  Prop osition 4.5. If E 6 = T and T is tota l ly r amiﬁe d over R , then Σ τ = E ∗ , so SK 1 ( E , τ ) = 1 . Pr o of. Th e case where E 0 = T 0 w as co vered by Prop. 4.4 . Thus, w e ma y assume that E 0 % T 0 . By Lemma 4.2 and Remark 4. 1 ( ii ), we can a ss u me that τ | E 0 is of the ﬁ rst kin d. F urther, we can assume that E ∗ 0 = Σ τ | E 0 ( E 0 ) . F or, if τ | E 0 is symplectic, tak e a ny a ∈ E ∗ 0 with τ ( a ) = − a , and let τ ′ = τ ϕ a . Then, τ ′ ∼ τ (see Remark 4.1 ( iii )). Also, τ ′ | Z ( E 0 ) = τ | Z ( E 0 ) , as a ∈ E 0 and so ϕ a | Z ( E 0 ) = id. Therefore, τ ′ | E 0 is of the ﬁrst kind. But as τ ( a ) = − a , τ ′ | E 0 is orthogonal. Thus E ∗ 0 = Σ τ ′ | E 0 ( E 0 ), as n oted at the b eginning of § 2.2.1 . No w r eplace τ by τ ′ . W e consider t wo cases. Case I. Sup p ose for eac h γ ∈ Γ E there is x γ ∈ E ∗ γ suc h that τ ( x γ ) = x γ . Th en, E ∗ = S γ ∈ Γ E E ∗ 0 x γ ⊆ Σ τ ( E ), as desired. Case I I. Su pp ose there is γ ∈ Γ E with E γ ∩ S τ = 0. Then τ ( d ) = − d for eac h d ∈ E γ . Fix t ∈ E ∗ γ . F or an y a ∈ E 0 , w e h a v e ta ∈ E γ ; so, − ta = τ ( ta ) = τ ( a ) τ ( t ) = − τ ( a ) t . T h at is, τ ( a ) = ϕ t ( a ) for all a ∈ E 0 . (4.1) Let τ ′′ = τ ϕ t , which is a u nitary in volutio n on E with τ ′′ ∼ τ (see Rema r k 4.1 ( iii )). But, τ ′′ ( a ) = a for all a ∈ E 0 , i.e., τ ′′ | E 0 = id. This imp lies that E 0 is a ﬁeld. Replace τ b y τ ′′ . Th e rest of the argumen t us es this new τ . So τ | E 0 = id. If we are no w in Case I for t h is τ , then we are done b y Case I. So, assume we are in C ase I I. T ak e any γ ∈ Γ E with E γ ∩ S τ = 0 . F or any nonzero t ∈ E γ , equation ( 4.1 ) app lies to t , sho wing ϕ t ( a ) = τ ( a ) = a for all a ∈ E 0 ; hence for the map Θ E of ( 2.3 ), Θ E ( γ ) = id E 0 . But recall that E 0 is 16 R. H AZRA T AND A. R. W AD SW OR TH Galois o v er T 0 and Θ E : Γ E → Gal( E 0 / T 0 ) is surjectiv e. Sin ce E 0 6 = T 0 , there is δ ∈ Γ E with Θ E ( δ ) 6 = id . Hence, th ere m u st b e some s ∈ E ∗ δ ∩ S τ . Like w ise, since Θ E ( γ − δ ) = Θ E ( γ )Θ E ( δ ) − 1 6 = id, there is some r ∈ E ∗ γ − δ ∩ S τ . Th en, as r s ∈ E ∗ γ , w e ha ve E ∗ γ = E ∗ 0 r s ⊆ Σ τ . This is true for ev ery γ w ith E γ ∩ S τ = 0. But for an y ot h er γ ∈ Γ E , there is an x γ in E ∗ γ ∩ S τ ; then E ∗ γ = E ∗ 0 x γ ⊆ Σ τ . Thus, E ∗ = S γ ∈ Γ E E ∗ γ ⊆ Σ τ .  4.2. T / R unramiﬁed. Let E b e a graded d ivision algebra with a u nitary in vol u tion τ suc h that T = Z ( E ) is unr amiﬁ ed o ver R = T τ . In this subs ection, w e w ill giv e a general formula for SK 1 ( E , τ ) in terms of data in E 0 . Lemma 4.6. Supp ose T is unr amiﬁe d over R . Then, (i) Every E γ c ontains b oth nonzer o symmetric and skew symmetric elements. (ii) Z ( E 0 ) is a gener alize d dihe dr al extension for T 0 over R 0 ( se e Def. 2.4 ) . (iii) If T is unr amiﬁe d over R , then SK 1 ( E , τ ) = Σ ′ 0 / Σ 0 . Pr o of. (i) If c har ( E ) = 2, it is easy to see that ev ery E γ con tains a symmetric element (wh ic h is also skew symmetric) regardless of any assu mption on T / R . Let char( E ) 6 = 2. Since [ T 0 : R 0 ] = 2 and R 0 = T τ 0 , there is c ∈ T 0 with τ ( c ) = − c . No w there is t ∈ E γ , t 6 = 0, with τ ( t ) = ǫt where ǫ = ± 1. Then τ ( ct ) = − ǫct . (ii) Let G = Gal( Z ( E 0 ) / R 0 ) and H = Gal( Z ( E 0 ) / T 0 ). Note that [ G : H ] = 2. Sin ce τ is unitary , τ | Z ( E 0 ) ∈ G \ H . W e w ill denote τ | Z ( E 0 ) b y τ and will s ho w th at for an y h ∈ H , ( τ h ) 2 = 1. By ( 2.3 ), Θ E : Γ E → Gal( Z ( E 0 ) / T 0 ) is on to, so there is γ ∈ Γ E , suc h that Θ E ( γ ) = h . Also by ( i ), ther e is an x ∈ E ∗ γ with τ ( x ) = x . Then τ ϕ x is an inv olution, where ϕ x is conjugation by x ; therefore, τ ϕ x | Z ( E 0 ) ∈ G has order 2. But ϕ x | Z ( E 0 ) = Θ E ( γ ) = h . Thus ( τ h ) 2 = 1. (iii) By ( i ), for eac h γ ∈ Γ E , there is s γ ∈ E γ , s γ 6 = 0, with τ ( s γ ) = s γ . B y Remark 4.1 ( iv ), Σ ′ τ = S γ ∈ Γ E s γ Σ ′ 0 . S in ce eac h s γ ∈ S τ ⊆ Σ τ , the injectiv e map Σ ′ 0 / Σ 0 → Σ ′ τ / Σ τ is an isomorphism.  T o simplify notation in the next theorem, let τ = τ | Z ( E 0 ) ∈ Gal( Z ( E 0 ) / R 0 ), and for an y h ∈ Gal( Z ( E 0 ) / T 0 ), write Σ hτ ( E 0 ) for Σ ρ ( E 0 ) for any unitary inv olution ρ on E 0 suc h that ρ | Z ( E 0 ) = hτ . This is well-deﬁned, indep end en t of the c hoice o f ρ , b y the un graded analogue of Remark( 4.1 )( ii ). Theorem 4.7. L et E b e a gr ade d division algebr a with c enter T , with a unitary gr ade d inv olution τ , such that T is unr amiﬁe d over R = T τ . F or e ach γ ∈ Γ E cho ose a nonzer o x γ ∈ S τ ∩ E γ . L et H = Gal( Z ( E 0 ) / T 0 ) . Then, SK 1 ( E , τ ) ∼ = (Σ ′ τ ∩ E 0 )  (Σ τ ∩ E 0 ) , with Σ ′ τ ∩ E 0 =  a ∈ E ∗ 0 | N Z ( E 0 ) / T 0 Nrd E 0 ( a ) ∂ ∈ R 0  , wher e ∂ = ind( E ) /  ind( E 0 ) [ Z ( E 0 ) : T 0 ]  (4.2) and Σ τ ∩ E 0 = P · X , wher e P = Q h ∈ H Σ h τ ( E 0 ) a nd X = h x γ x δ x − 1 γ + δ | γ , δ ∈ Γ E i ⊆ E ∗ 0 . (4.3) F urthermor e, if H = h h 1 , . . . , h m i , then P = Q ( ε 1 ,...,ε m ) ∈{ 0 , 1 } m Σ h ε 1 1 ...h ε m m τ ( E 0 ) . Before pro vin g th e theorem, we record the follo wing: Lemma 4.8 . L et A b e a c entr al simple algebr a o ve r a ﬁeld K , with an involution τ and an automo rphism or a nti- automorp hism σ . Then, (i) σ τ σ − 1 is an involution of A of the sam e kind as τ , and S στ σ − 1 = σ ( S τ ) , so Σ στ σ − 1 = σ (Σ τ ) . UNIT AR Y SK 1 OF DIVISION ALGEBRAS 17 (ii) Supp ose A is a division ring. If σ and τ ar e e ach unitar y involutions, then ( wr iting S ∗ τ = S τ ∩ A ∗ ) , S ∗ τ ⊆ S ∗ σ · σ ( S ∗ τ ) = S ∗ σ · S ∗ στ σ − 1 , so Σ τ ⊆ Σ σ · Σ στ σ − 1 . Pr o of. (i) This follo ws b y easy calculations. (ii) Obser ve that if a ∈ S ∗ τ , then a =  aσ ( a )  σ ( a − 1 ) with aσ ( a ) ∈ S ∗ σ and σ ( a − 1 ) ∈ σ ( S ∗ τ ) = S ∗ στ σ − 1 b y ( i ). T h u s , ( ii ) fol lows from ( i ) and the f act that A ′ ⊆ Σ τ ∩ Σ σ (see ( 2.10 )).  Pr o of of The or em 4.7 . First note that by Lemma 4.6 ( iii ) the c anonical map (Σ ′ τ ∩ E 0 )  (Σ τ ∩ E 0 ) − → Σ ′ τ / Σ τ = SK 1 ( E , τ ) is an isomorphism. The description of Σ ′ τ ∩ E 0 in ( 4.2 ) is immediate from the fact that for a ∈ E 0 , Nrd E ( a ) = N Z ( E 0 ) / T 0 Nrd E 0 ( a ) ∂ ∈ T 0 (see Remark 2.1 ( iii )). F or Σ τ ∩ E 0 , note that for eac h γ ∈ Γ E , if a ∈ E 0 , then ax γ ∈ S τ if and only if x γ τ ( a ) x − 1 γ = a . Th at is, S τ ∩ E γ = S ( ϕ x γ τ ; E 0 ) x γ , where S ( ϕ x γ τ ; E 0 ) denotes the set of sym metric el ements in E 0 for the unitary in vol u tion ϕ x γ τ | E 0 . Th erefore, Σ τ ∩ E 0 =  S ( ϕ x γ τ ; E 0 ) ∗ x γ | γ ∈ Γ E  ∩ E 0 . T ak e a p ro duct a 1 x 1 . . . a k x k in Σ τ ∩ E 0 where eac h x i = x γ i for s ome γ i ∈ Γ E and a i ∈ S ( ϕ x i τ ; E 0 ) ∗ . Then, a 1 x 1 . . . a k x k = a 1 ϕ x 1 ( a 2 ) . . . ϕ x 1 ...x i − 1 ( a i ) . . . ϕ x 1 ...x k − 1 ( a k ) x 1 . . . x k ∈ E γ 1 + ... + γ k . (4.4) So, γ 1 + . . . + γ k = 0. No w, as a i ∈ S ( ϕ x i τ ; E 0 ) and τ ϕ − 1 x j = ϕ x j τ for a ll j , by Lemma 4.8 ( i ) w e o b tain ϕ x 1 ...x i − 1 ( a i ) ∈ S ( ϕ x 1 . . . ϕ x i − 1 ( ϕ x i τ ) ϕ − 1 x i − 1 . . . ϕ − 1 x 1 ; E 0 ) ∗ = S ( ϕ x 1 ...x i − 1 x i x i − 1 ...x 1 τ ; E 0 ) ∗ ⊆ Σ h τ ( E 0 ) ⊆ P , (4.5) where h = ϕ x 1 ...x i − 1 x i x i − 1 ...x 1 | Z ( E 0 ) ∈ H . Note also that if k = 1, then x 1 ∈ S τ ∩ E ∗ 0 ⊆ Σ τ ( E 0 ) ⊆ P . If k > 1, then x 1 . . . x k = x γ 1 . . . x γ k = ( x γ 1 x γ 2 x − 1 γ 1 + γ 2 )( x γ 1 + γ 2 x γ 3 . . . x γ k ) , with ( γ 1 + γ 2 ) + γ 3 + . . . + γ k = 0. It follo ws by induction on k that x 1 . . . x k ∈ X . With this and ( 4.4 ) and ( 4.5 ), w e ha ve a 1 x 1 . . . a k x k ∈ P · X (whic h is a group, as E ′ 0 ⊆ Σ τ ( E 0 ) ⊆ P b y ( 2.10 )), sho wing that Σ τ ∩ E 0 ⊆ P · X . F or the reve r se inclusion, take an y h ∈ H and choose γ ∈ Γ E with ϕ x γ | Z ( E 0 ) = h . Then, x γ ∈ S ∗ τ ⊆ Σ τ and S ( ϕ x γ τ ; E 0 ) ∗ x γ = S ∗ τ ∩ E γ ⊆ Σ τ , so Σ h τ ( E 0 ) = Σ ϕ x γ τ ( E 0 ) = h S ( ϕ x γ τ ; E 0 ) ∗ i ⊆ Σ τ ∩ E 0 . Th u s, P ⊆ Σ τ ∩ E 0 , and clearly also X ⊆ Σ τ ∩ E 0 . Hence, Σ τ ∩ E 0 = P · X . The ﬁnal equalit y for P in the T heorem follo ws fr om Lemma 4.9 b elo w b y taking U = E ∗ 0 , A = H , and W h = Σ h τ ( E 0 ) for h ∈ H . T o see that the lemma applies, note that eac h Σ hτ ( E 0 ) con tains E ′ 0 b y ( 2.10 ). F urthermore, ta ke any h, ℓ ∈ H , and c h o ose x, y ∈ E ∗ ∩ S τ with ϕ x | Z ( E 0 ) = h and ϕ y | Z ( E 0 ) = ℓ . Then, ( ϕ y τ )( ϕ x τ )( ϕ y τ ) − 1 = ϕ y τ ϕ x ϕ − 1 y = ϕ y x − 1 y τ , and ϕ y x − 1 y | Z ( E 0 ) = ℓ h − 1 ℓ = ℓ 2 h − 1 . Hence, by Lemma 4.8 ( ii ), Σ h τ ( E 0 ) ⊆ Σ ℓ τ ( E 0 )Σ ℓ 2 h − 1 τ ( E 0 ). This sho ws that hypothesis ( 4.6 ) of Lemma 4.9 belo w is satisﬁed h ere.  Lemma 4.9. L et U b e a gr oup, A an ab elian gr oup, and { W a | a ∈ A } a family of sub gr oups of U with e ach W a ⊇ [ U, U ] . Supp ose W a ⊆ W b W 2 b − a for al l a, b ∈ A. (4.6) If A = h a 1 , . . . , a m i , then Q a ∈ A W a = Q ( ε 1 ,...,ε m ) ∈{ 0 , 1 } m W ε 1 a 1 + ... + ε m a m . 18 R. H AZRA T AND A. R. W AD SW OR TH Pr o of. S ince eac h W a ⊇ [ U, U ], w e ha ve W a W b = W b W a , and this is a subgroup of U , for a ll a, b ∈ A . Let Q = Q ( ε 1 ,...,ε m ) ∈{ 0 , 1 } m W ε 1 a 1 + ... + ε m a m . W e pro v e b y in d uction on m that eac h W a ⊆ Q . Th e lemma then follo ws, as Q is a subgroup of U . Note that condition ( 4.6 ) can b e conv en iently restated, if a + b = 2 d ∈ A, th en W a ⊆ W d W b . (4.7) T ak e an y c ∈ A . Th en, ( 4.7 ) shows that W − c ⊆ W 0 W c . T ak e an y i ∈ Z , and supp ose W ic ⊆ W 0 W c . Th en by ( 4.7 ) W − ic ⊆ W 0 W ic ⊆ W 0 W c . So, by ( 4.7 ) again, W ( i +2) c ⊆ W c W − ic ⊆ W 0 W c and W ( i − 2) c ⊆ W − c W − ic ⊆ W 0 W c . Hence, by induction (starting with j = 0 and j = 1), W j c ⊆ W 0 W c for every j ∈ Z . T h is p ro ve s the lemma when m = 1. No w assu me m > 1 and let B = h a 1 , . . . , a m − 1 i ⊆ A . By in d uction, for all b ∈ B , W b ⊆ Q ( ε 1 ,...,ε m − 1 ) ∈{ 0 , 1 } m − 1 W ε 1 a 1 + ... + ε m − 1 a m − 1 ⊆ Q. Also, b y the cyclic ca se d one ab o v e, W j a m ⊆ W 0 W a m ⊆ Q for all j ∈ Z . S o, for any b ∈ B , j ∈ Z , using ( 4.7 ), W 2 b + j a m ⊆ W b W − j a m ⊆ Q, (4.8) and W b +2 j a m ⊆ W j a m W − b ⊆ Q. (4.9) Let d = a i 1 + . . . + a i ℓ for an y indices 1 ≤ i 1 < i 2 < . . . < i ℓ ≤ m − 1. Since W d + a m ⊆ Q b y h yp othesis, from ( 4.7 ) and ( 4.9 ) it fol lows that W d +3 a m ⊆ W d +2 a m W d + a m ⊆ Q. (4.10) No w, tak e an y element of A ; it has the form b + j a m for some b ∈ B and j ∈ Z . If b ∈ 2 B or if j is ev en, then ( 4.8 ) and ( 4.9 ) show that W b + j a m ⊆ Q . The remaining case is that j is o dd and b 6∈ 2 B , so b = 2 c + d , where c ∈ B and d = a i 1 + . . . + a i ℓ for some indices with 1 ≤ i 1 < i 2 < . . . < i ℓ ≤ m − 1. Set q = 1 if j ≡ 3 (mo d 4) and q = 3 if j ≡ 1 (mod 4 ). T hen, W d + q a m ⊆ Q b y deﬁn ition if q = 1 or b y ( 4.10 ) if q = 3. Hence, b y ( 4.7 ), W b + j a m = W (2 c + d )+ j a m ⊆ W ( c + d )+(( j + q ) / 2) a m W d + q a m ⊆ Q, (4.11) using ( 4.9 ) as c + d ∈ B and ( j + q ) / 2 is eve n . Thus, W a ⊆ Q , for all a ∈ A .  Corollary 4.10. If E is unr amiﬁe d over R , then SK 1 ( E , τ ) ∼ = SK 1 ( E 0 , τ | E 0 ) . Pr o of. S ince E is u nramiﬁed o v er R , we ha v e T is un ramiﬁed o v er R , Z ( E 0 ) = T 0 , and Γ E = Γ R , so w e ca n c ho ose all the x γ ’s to lie in R . T h e assertion thus follo ws immediately from Th. 4.7 , as P = Σ τ | E 0 ( E 0 ) and X ⊆ R ∗ 0 ⊆ P . (Alt ern ativ ely , more directly , one ca n observ e that Σ ′ 0 = Σ ′ τ | E 0 ( E 0 ) and Σ 0 = Σ τ | E 0 ( E 0 ) and so deduce the Corollary by L emma 4.6 ( iii ).)  Corollary 4.11. If T is unr amiﬁe d over R and E has a maximal gr ade d subﬁeld M u nr amiﬁe d over T and another maximal gr ade d subﬁeld L total ly r amiﬁe d over T with τ ( L ) = L , then E is semir amiﬁe d with E 0 = M 0 ( a ﬁeld ) and Γ E = Γ L , a nd SK 1 ( E , τ ) ∼ =  a ∈ E 0 | N E 0 / T 0 ( a ) ∈ R 0   Q h ∈ Gal( E 0 / T 0 ) E ∗ h τ 0 . (4.12) Pr o of. Let n = ind( E ). Since [ M 0 : T 0 ] = n and | Γ L : Γ T | = n , it follo ws from the F und amen tal Eq u al- it y ( 2.2 ) for E / T , M / T , and L / T that [ E 0 : T 0 ] = [Γ E : Γ T ] = n , E 0 = M 0 , which is is a ﬁeld, and Γ E = Γ L . Th u s E is semir amiﬁed, so for the ∂ of ( 2.6 ), ∂ = 1. No w, L τ is a graded su bﬁeld of L with [ L : L τ ] = 2. Since L 0 = T 0 while ( L τ ) 0 = ( L 0 ) τ = R 0 , L m ust b e u nramiﬁed o ver L τ ; hence, Γ E = Γ L = Γ L τ . Th erefore, UNIT AR Y SK 1 OF DIVISION ALGEBRAS 19 one can choose all the x γ ’s in T h. 4.7 to lie in L τ . Then eac h x γ x δ x − 1 γ + δ ∈ ( L τ ) ∗ 0 = R ∗ 0 = E ∗ τ 0 . Hence, X ⊆ P = Q h ∈ Gal( E 0 / T 0 ) E ∗ h τ 0 , so the formula for SK 1 ( E , τ ) in Th. 4.7 reduces to ( 4.12 ).  R emark 4.12 . In a sequel to this pap er [ W 3 ], the follo wing w ill b e sho w n: With the h yp otheses of Th. 4.7 , supp ose E is semiramiﬁed with a graded maximal subﬁeld L totally ramiﬁed o ve r T suc h that τ ( L ) = L , and supp ose Gal( E 0 / T 0 ) is bicyclic, say E 0 = N ⊗ T 0 N ′ with N and N ′ eac h cyclic Galois o ver T 0 . Th en, SK 1 ( E , τ ) ∼ = Br( E 0 / T 0 ; R 0 )  Br( N/ T 0 ; R 0 ) + Br( N ′ / T 0 ; R 0 )  , where Br( E 0 / T 0 ) is the relativ e Br au er group ker  Br( T 0 ) → Br( E 0 )  and Br( E 0 / T 0 ; R 0 ) is the k ern el of cor E 0 → R 0 : Br( E 0 / T 0 ) → Br( E 0 / R 0 ). (Compare th is with [ Y 2 , T h . 5.6].) A fu r ther formula will b e giv en assuming only that E is semir amiﬁed o v er T with Gal( E 0 / T 0 ) bicyclic. In his construction of d ivision algebras D with nontrivial SK 1 , Platono v work ed originally in [ P 2 , § 4] with a division a lgebra D wh er e Z ( D ) is a Lauren t pow er series ﬁeld; he ga ve an exact sequence relating SK 1 ( D ) with S K 1 ( D ) and what he called the “group of p ro jectiv e conorms.” Y anc h evski ˘ ı ga ve in [ Y 2 , 4.11] a n analogous exact sequ ence in the unitary case. T heir results and proofs are v alid whenev er Z ( D ) has a henselian discrete (r an k 1) v aluation. W e sh o w here that their results hold more generally wh enev er Z ( D ) has a henselian v aluation with Γ D / Γ Z ( D ) cyclic. W e wo rk in the equiv alen t graded setting where the argumen ts are more transp aren t. As b efore, let E b e a graded d ivision algebra ﬁnite d im en sional ov er its cent er T with a unitary grad ed in vol u tion τ , and let R = T τ . Assume that T is unr amiﬁed ov er R and that Γ E / Γ T is cyclic group. (This cyclicit y h olds , e.g., wheneve r Γ T ∼ = Z .) It follo w s that the surjectiv e map Θ E : Γ E → Gal( Z ( E 0 ) / T 0 ) has k ernel Γ T . (F or, b y [ HW 2 , Prop. 2.1, (2.3) , Remark 2.4(i)] , k er (Θ E ) / Γ T has a n ondegenerate sym plectic pairing, and hence has ev en rank as a ﬁnite ab elian group. But here ke r(Θ E ) / Γ T is a cyclic group.) Hence, ∂ = 1 b y Lemma 2.2 , so E is inertially sp lit. In voking Lemma 4.6 ( i ), c h o ose an y s ∈ E ∗ with deg ( s ) + Γ T a generator of Γ E / Γ T , such that τ ( s ) = s . Let σ = ϕ s ∈ Aut T ( E ); so σ τ is a n other T / R -graded inv olution of E , and τ σ = σ − 1 τ (see Rema r k 4.1 ( iii )). By the c hoice of s , σ | Z ( E 0 ) is a generator of the cyclic group Gal( Z ( E 0 ) / T 0 ). Note that Gal( Z ( E 0 ) / R 0 ) = h σ | Z ( E 0 ) , τ | Z ( E 0 ) i is a dihedral group. Recall our con ven tion that c στ means σ ( τ ( c )). Let S =  ( β , b ) ∈ Z ( E 0 ) ∗ × E ∗ 0 | β σ − 1 = Nrd E 0 ( b )  ; N = π 1 ( S ) (pro jection in to the ﬁr st comp onen t) =  β ∈ Z ( E 0 ) ∗ | β σ − 1 = Nrd E 0 ( b ) for some b ∈ E ∗ 0  ; W = T ∗ 0 · Nrd E 0 ( E ∗ 0 ) ⊆ Z ( E 0 ) ∗ ; P = N / W , w h ic h is Pla tonov’s group of pro jectiv e co n orms for E [ P 2 , § 4] . S τ =  ( α, a ) ∈ Z ( E 0 ) ∗ × E ∗ 0 | α σ − 1 = Nrd E 0 ( a ) 1 − στ  ; N τ = π 1 ( S τ ) (pro jection int o the ﬁ rst comp onent ) =  α ∈ Z ( E 0 ) ∗ | α σ − 1 = Nrd E 0 ( a ) 1 − στ for some a ∈ E ∗ 0  ; W τ = T ∗ 0 · Nrd E 0 (Σ τ ( E 0 )) ⊆ Z ( E 0 ) ∗ ; P U τ = N τ / W τ , which is Y anc h evski ˘ ı’s group of unitary pr o jectiv e conorms for ( E , τ ) [ Y 2 , 4.11] . Prop osition 4.13. If T is unr amiﬁe d over R and Γ E / Γ T is cyclic, then for any gener ator σ of the cyclic gr oup Gal( Z ( E 0 ) / T 0 ) , we have (i) SK 1 ( E ) ∼ = { a ∈ E ∗ 0 | N Z ( E 0 ) / T 0 (Nrd E 0 ( a )) = 1 }   [ E ∗ 0 , E ∗ 0 ] · { c σ − 1 | c ∈ E ∗ 0 }  . (ii) SK 1 ( E , τ ) ∼ = { a ∈ E ∗ 0 | N Z ( E 0 ) / T 0 (Nrd E 0 ( a )) ∈ R 0 }   Σ τ ( E 0 ) · Σ στ ( E 0 )  . 20 R. H AZRA T AND A. R. W AD SW OR TH (iii) The fol lowing se quenc e is exa ct: SK 1 ( E 0 , σ τ ) − → SK 1 ( E , τ ) f − → P U τ − → 1 , (4.1 3) wher e the map f : SK 1 ( E , τ ) → P U τ is the c omp osition of a Σ τ ( E 0 ) · Σ στ ( E 0 ) 7→ ( α, a ) ∈ S τ and ( α, a ) 7→ α W τ ∈ P U τ . (iv) Ther e is a c ommutative diagr am with exact r ows: SK 1 ( E 0 , σ τ ) / / a ↓ a 1 − στ   SK 1 ( E , τ ) f / / a ↓ a 1 − στ   P U τ / / α ↓ α   1 1 / / SK 1 ( E 0 ) / / b ↓ b   SK 1 ( E ) g / / b ↓ b   P / / β ↓ β 1+ τ   1 SK 1 ( E 0 , σ τ ) / / SK 1 ( E , τ ) / / P U τ / / 1 wher e the map g : SK 1 ( E ) → P is the c omp osition of b [ E ∗ 0 , E ∗ 0 ] h c σ − 1 i 7→ ( β , b ) ∈ S and ( β , b ) 7→ β W ∈ P . (v) If E 0 is a ﬁeld, then SK 1 ( E , τ ) = 1 . Pr o of. (i) This form u la w as giv en b y Suslin [ S 1 , Prop. 1.7] for a division a lgebra o ver a ﬁeld with a complete discrete v aluation. In ord er to p ro v e it in the graded setting we need t w o exact sequences whic h we re giv en in [ HaW , Th. 3.4]: Γ E / Γ T ∧ Γ E / Γ T − → E (1)  [ E ∗ 0 , E ∗ ] − → SK 1 ( E ) − → 1 , 1 − → k er e N / [ E ∗ 0 , E ∗ ] − → E (1)  [ E ∗ 0 , E ∗ ] − → µ ∂ ( T 0 ) ∩ e N ( E ∗ 0 ) − → 1 , where E (1) = { a ∈ E ∗ | Nrd E ( a ) = 1 } ⊆ E 0 and e N = N Z ( E 0 ) / T 0 ◦ Nr d E 0 : E ∗ 0 → T ∗ 0 . Since ∂ = 1 (see the paragraph prior to the Prop osition) and the w ed ge p ro duct of a cyclic group with itself is trivial, these exact sequences yield SK 1 ( E ) ∼ = { a ∈ E ∗ 0 | N Z ( E 0 ) / T 0 (Nrd E 0 ( a )) = 1 }  [ E ∗ 0 , E ∗ ] . W e are left to sho w that [ E ∗ 0 , E ∗ ] = [ E ∗ 0 , E ∗ 0 ] ·{ c σ − 1 | c ∈ E ∗ 0 } . Th is follo ws from the fact th at E ∗ / T ∗ E ∗ 0 ∼ = Γ E / Γ T is cyclic together with the follo win g observ ation, w hic h is easily v eriﬁed using the standard comm utator iden tities: If G is a group and N is a normal subgroup of G such that G/ Z ( G ) N is a cyclic group generated b y , say , xZ ( G ) N , th en [ N , G ] = [ N , N ][ x, N ] where [ x, N ] = { [ x, n ] | n ∈ N } . (Here, tak e G = E ∗ , N = E ∗ 0 , and for x tak e an y s ∈ E ∗ γ for an y γ ∈ Γ E suc h th at Θ E ( γ + Γ T ) = σ .) (ii) By Th. 4.7 , taking into accoun t that ∂ = 1 and Gal ( Z ( E 0 ) / T 0 ) = h σ i , we ha v e, SK 1 ( E , τ ) ∼ = (Σ ′ τ ∩ E 0 )  (Σ τ ∩ E 0 ) = { a ∈ E ∗ 0 | N Z ( E 0 ) / T 0 Nrd E 0 ( a ) ∈ R 0 }   Σ τ ( E 0 ) · Σ στ ( E 0 ) · h x γ x δ x − 1 γ + δ | γ , δ ∈ Γ E i  , (4.14) where for eac h γ ∈ Γ E , x γ is c hosen in E ∗ γ with x γ = τ ( x γ ) and x γ 6 = 0, us ing Lemma 4.6 ( i ). Let L = R [ s ], where s is c h osen in E ∗ with ϕ ( s ) | E 0 = σ , whic h is p ossible a s Θ E : Γ E → Gal( Z ( E 0 ) / T 0 ) is surjectiv e (see ( 2.3 )). Moreo v er, s ca n b e chosen with τ ( s ) = s . Since k er (Θ E ) = Γ T = Γ R , we ha ve Γ E = h deg( s ) i + Γ R . Th u s, L is a graded su bﬁeld of E with Γ L = Γ E and τ | L = id. F or eac h γ ∈ Γ E w e can c ho ose x γ ∈ L ∗ γ ; then for all γ , δ ∈ Γ E , we h a v e x γ x δ x − 1 γ + δ ∈ L ∗ 0 ⊆ Σ τ ( E 0 ). Thus, the h x γ x δ x − 1 γ + δ i term in ( 4.14 ) is r edundant, yielding the formula in ( ii). (iii) W e ﬁrst c hec k that f is w ell-deﬁned: T ake any a ∈ E ∗ 0 with N Z ( E 0 ) / T 0 (Nrd E 0 ( a )) ∈ R ∗ 0 . Let c = Nrd E 0 ( a ). Th en, as R 0 = T τ 0 , 1 = N Z ( E 0 ) / T 0 ( c ) 1 − τ = N Z ( E 0 ) / T 0 ( c ) 1 − στ = N Z ( E 0 ) / T 0 ( c 1 − στ ). By Hilb ert 90, there is α ∈ Z ( E 0 ) ∗ with α σ − 1 = c 1 − στ = Nrd E 0 ( a ) 1 − στ . Hence ( α, a ) ∈ S τ , so α ∈ N τ , and UNIT AR Y SK 1 OF DIVISION ALGEBRAS 21 the choice of α is unique up to T ∗ 0 ⊆ W τ . Thus, the image of a in P U τ is indep endent of the choic e of α . Supp ose fur ther that a = pq for some p ∈ Σ τ ( E 0 ), q ∈ Σ στ ( E 0 ), say , p = s 1 . . . s k with eac h s i ∈ S τ ( E 0 ). Then, Nrd E 0 ( p ) τ = Nrd E 0 ( s 1 ) τ . . . Nrd E 0 ( s k ) τ = Nrd E 0 ( s τ 1 ) . . . Nrd E 0 ( s τ k ) = Nrd E 0 ( s 1 ) . . . Nrd E 0 ( s k ) = Nrd E 0 ( p ); (4.15) lik ewise, Nrd E 0 ( q ) στ = Nrd E 0 ( q ). S o, α σ − 1 = Nrd E 0 ( pq ) 1 − στ = Nrd E 0 ( p ) 1 − στ Nrd E 0 ( q ) 1 − στ = Nrd E 0 ( p ) 1 − σ . Hence,  α Nrd E 0 ( p )  σ − 1 = 1, showing th at α Nrd E 0 ( p ) ∈ T 0 ; Thus α ∈ W τ . This p ro ve s that f is w ell- deﬁned. F or the sub jectivit y of f , tak e an y α ∈ N τ . Then, there is a ∈ E ∗ 0 with α σ − 1 = Nrd E 0 ( a ) 1 − στ . So, N Z ( E 0 ) / T 0 (Nrd E 0 ( a )) 1 − στ = N Z ( E 0 ) / T 0 ( α σ − 1 ) = 1, whic h sho ws that N Z ( E 0 ) / T 0 (Nrd E 0 ( a )) ∈ T στ 0 = T τ 0 = R 0 , and hence a ∈ Σ ′ τ ( E ) ∩ E ∗ 0 . S in ce f  a Σ τ ( E 0 )Σ στ ( E 0 )  = α W τ , f is sur jectiv e. Finally , we determine k er ( f ): The image of SK 1 ( E 0 , σ τ ) in S K 1 ( E , τ ) is Σ ′ στ ( E 0 )Σ τ ( E 0 )  Σ στ ( E 0 )Σ τ ( E 0 ). An elemen t in this image is represented b y some a ∈ Σ ′ στ ( E 0 ). F or suc h an a , Nrd E 0 ( a ) 1 − στ = 1. T hen (1 , a ) ∈ S τ , so that f m aps the image of a to 1 in P U τ . Con versely , sup p ose a Σ τ ( E 0 )Σ στ ( E 0 ) ∈ k er ( f ). That is, Nrd E 0 ( a ) 1 − στ = α σ − 1 , where α ∈ W τ , so α = c Nrd E 0 ( d ) w ith c ∈ T ∗ 0 and d ∈ Σ τ ( E 0 ). So, Nrd E 0 ( d ) = Nrd E 0 ( d ) τ b y the argumen t of ( 4.15 ) ab o ve, and hence Nrd E 0 ( a ) 1 − στ = α σ − 1 =  c Nrd E 0 ( d )  σ − 1 = Nrd E 0 ( d ) σ − 1 = Nrd E 0 ( d ) στ − 1 . Th u s, Nrd E 0 ( ad ) 1 − στ = 1, i.e., ad ∈ Σ ′ στ ( E 0 ). Hence, a = ( ad ) d − 1 ∈ Σ ′ στ ( E 0 )Σ τ ( E 0 ). T his shows that k er( f ) coincides with th e imag e of S K 1 ( E 0 , σ τ ) in S K 1 ( E , τ ), completing the pr o of of exactness of th e sequence. (iv) Exactness of the middle ro w is pro ved b y an analogo us but easier argumen t to that for ( iii ). Comm utativit y of the left rectangles of th e diagram is evident. Comm utativit y of th e top r igh t rectangle is clear from the deﬁ n itions. Comm u tativit y of the b ottom right rectangle is easy to c heck using the identit y (1 − σ τ ) ◦ ( σ − 1) = ( σ − 1) ◦ (1 + τ ) , (4. 16) whic h follo ws f r om ( σ τ ) 2 = id. Note that f or eac h column of the diagram, the comp osition of th e t wo maps is the squaring map. (v) F or this part, the pro of follo ws closely Y anc hevski ˘ ı’s pro of in [ Y 2 , 4.13]. (But our notational con v ention for pro d ucts of fu nctions is f g = f ◦ g , wh ereas h is app ears to b e f g = g ◦ f .) Supp ose E 0 is a ﬁeld. F or simplicit y we denote τ = τ | E 0 b y τ . T ak e a ∈ Σ ′ τ ( E ) ∩ E 0 . S o, N E 0 / T 0 ( a 1 − τ ) = 1. W e will s h o w that a ∈ E τ 0 E στ 0 . It then follo ws by ( ii ) a b o v e that S K 1 ( E , τ ) = 1. But since E 0 is cyclic o ve r T 0 , b y Hilb ert 90 there is a b ∈ E ∗ 0 suc h that a τ − 1 = b σ − 1 where h σ i = Gal( E 0 / T 0 ). So, 1 = a ( τ +1)( τ − 1) = b ( τ +1)( σ − 1) . Analogously to ( 4.16 ), we h av e ( τ + 1)( σ − 1) = ( σ − 1)(1 − τ σ ). So b ( σ − 1)(1 − τ σ ) = 1. Setting c = b (1 − τ σ ) , w e h a v e c σ − 1 = 1, so c ∈ T 0 . But, N T 0 / R 0 ( c ) = c 1+ τ σ = b (1+ τ σ )(1 − τ σ ) = 1. By Hilb ert 90 we hav e c = d τ σ − 1 for some d ∈ T ∗ 0 . Let t = bd ∈ E ∗ 0 . Then, t 1 − τ σ = b (1 − τ σ ) d (1 − τ σ ) = d ( τ σ − 1) d (1 − τ σ ) = 1, i.e., t ∈ E τ σ 0 . So, σ ( t ) = τ ( t ) ∈ E στ 0 . Th us, a τ − 1 = b σ − 1 = ( t/d ) σ − 1 = t σ − 1 = t τ − 1 , as d ∈ T 0 . This sho w s that ( aτ ( t )) τ − 1 = 1, i.e., aτ ( t ) ∈ E τ ; hence a = ( aτ ( t )) τ ( t ) − 1 ∈ E τ 0 E στ 0 .  5. Tot all y ramified algebras F or a graded division algebra E totally ramiﬁed o v er its cent er T with a u nitary graded in volution τ , t wo p ossible cases can arise: either T is totally ramiﬁed o v er R = T τ , or T is unr amiﬁed o v er R . In the ﬁrst case, w e show ed in Prop. 4.4 that SK 1 ( E , τ ) is trivial. W e no w obtain an easily computable explicit 22 R. H AZRA T AND A. R. W AD SW OR TH form u la for SK 1 ( E , τ ) in the second case. F or a ﬁeld K and for n ∈ N , we write µ n for th e group of all n -th ro ots of unity in an algebraic closure of K . Th en s et µ n ( K ) = µ n ∩ K ∗ . Theorem 5.1. If E i s tot al ly r amiﬁe d over T of index n and T is unr amiﬁe d over R , th en SK 1 ( E , τ ) ∼ =  a ∈ T ∗ 0 | a n ∈ R ∗ 0 }  { a ∈ T ∗ 0 | a e ∈ R ∗ 0 } (5.1) ∼ =  ω ∈ µ n ( T 0 ) | τ ( ω ) = ω − 1  µ e , (5.2) wher e e is the exp onent of Γ E / Γ T . In p articular, (i) The r estriction of the map K 1 ( E , τ ) → K 1 ( E ) given by a Σ τ 7→ a 1 − τ E ′ , ind u c es an inje ctive map α : SK 1 ( E , τ ) − → SK 1 ( E ) ∼ = µ n ( T 0 ) /µ e . (ii) If the exp onent e o f E is o dd, then α is an isom orphism. (iii) If e > 2 then T 0 = R 0 ( µ e ) , and τ acts on µ e by ω 7→ ω − 1 . Pr o of. S ince T is unramiﬁed o ve r R and E 0 = T 0 , the formulas of Th. 4.7 for SK 1 ( E , τ ) redu ce to ∂ = n and SK 1 ( E , τ ) ∼ = { a ∈ T ∗ 0 | a n ∈ R ∗ 0 }   R ∗ 0 h x γ x δ x − 1 γ + δ | γ , δ ∈ Γ E i  , (5.3) where eac h x γ ∈ E ∗ γ with τ ( x γ ) = x γ . Recall that as E / T is tota lly ramiﬁed, the canonical pairing E ∗ × E ∗ → µ e ( T 0 ) give n b y ( s, t ) 7→ [ s , t ] is surjectiv e ([ HW 2 , Prop. 2.1]), and µ e ( T 0 ) = µ e , i.e., T 0 con tains all e -th ro ots of unity . Sin ce eac h E γ = T 0 x γ with T 0 cen tral, it follo ws that { [ x δ , x γ ] | γ , δ ∈ Γ E } = µ e . No w consider c = x γ x δ x − 1 γ + δ for a ny γ , δ ∈ Γ E . Then, τ ( c ) = x − 1 γ + δ x δ x γ . Note that x δ x γ and x γ + δ eac h lie in E γ + δ = T 0 x γ + δ , so they commute. Hence, τ ( c ) c − 1 = x − 1 γ + δ ( x δ x γ ) x γ + δ x − 1 δ x − 1 γ = [ x δ , x γ ] . (5.4) Since [ x δ , x γ ] ∈ µ e , this sho w s that c ∈  a ∈ T ∗ 0 | a e ∈ R ∗ 0  . F or the rev erse inclusion, tak e any d in T ∗ 0 suc h that d e ∈ R ∗ 0 . So τ ( d ) d − 1 ∈ µ e . Thus, τ ( d ) d − 1 = [ x δ , x γ ], for some γ , δ ∈ Γ E . T aking c = x γ x δ x − 1 γ + δ , we hav e τ ( d ) d − 1 = τ ( c ) c − 1 b y ( 5.4 ), whic h implies that dc − 1 is τ -stable, so lies in R ∗ 0 ; th us, d ∈ R ∗ 0 h x γ x δ x − 1 γ + δ | γ , δ ∈ Γ E i . T herefore, R ∗ 0 h x γ x δ x − 1 γ + δ | γ , δ ∈ Γ E i = { a ∈ T ∗ 0 | a e ∈ R ∗ 0 } . Inserting this in ( 5.3 ) we obtain ( 5.1 ). (i) Consider the well- d eﬁned map α : SK 1 ( E , τ ) → SK 1 ( E ) giv en by a Σ τ 7→ a 1 − τ E ′ (see diagram ( 3 .9 ) for the non-graded v er s ion). By [ HaW , C or. 3.6(ii)], SK 1 ( E ) ∼ = µ n ( T 0 ) /µ e . T aking in to account form ula ( 5.1 ) for SK 1 ( E , τ ), it is easy to see that α is injectiv e. W e no w v erify that im( α ) =  ω ∈ µ n ( T 0 ) | τ ( ω ) = ω − 1   µ e , (5.5) and th us obtain ( 5.2 ). Indeed, since µ e = { [ x δ , x γ ] | γ , δ ∈ Γ E } , by setting c = x γ x δ x − 1 γ + δ w e hav e [ x δ , x γ ] = τ ( c ) c − 1 b y ( 5.4 ). This shows that µ e ⊆  ω ∈ µ n ( T 0 ) | τ ( ω ) = ω − 1  . No w for an y ω ∈ µ n ( T 0 ) with τ ( ω ) = ω − 1 , we hav e N T 0 / R 0 ( ω ) = 1, so Hi lb ert 90 guaran tees that ω = c 1 − τ for some c ∈ T ∗ 0 . Then, ( c n ) 1 − τ = ω n = 1, so c n ∈ R ∗ 0 . Thus, c ∈ Σ ′ τ , and clearly α ( c Σ τ ) = ω µ e . This s ho ws ⊇ in ( 5.5 ); the r ev erse inclusion is clear fr om the deﬁnition of α . (ii) Supp ose e is o dd. Let m = | µ n ( T 0 ) | . So, µ n ( T 0 ) = µ m , with m | n . Also, e | m , as µ e ⊆ T 0 . Since e and n h a v e the s ame prime factors, t h is is also tr u e for e and m . Recall that Aut( µ m ) ∼ = ( Z /m Z ) ∗ , the m ultiplicativ e group of units of the ring Z /m Z ; so, | Aut( µ m ) | = ϕ ( m ), where ϕ is Eu ler’s ϕ - fu nction. Since e | m and e and m hav e the same prime factors (all o dd), the canonical map ψ : Aut( µ m ) → Aut( µ e ) giv en b y restriction is surjectiv e with k ernel of order ϕ ( m ) /ϕ ( e ) = m/e , whic h is o d d. Therefore, ψ induces a n isomorphism on the 2-torsion subgrou p s, 2 Aut( µ m ) ∼ = 2 Aut( µ e ). No w , τ | µ m ∈ 2 Aut( µ m ) and we saw for (i) that τ | µ e is the inv erse map ω 7→ ω − 1 . The in verse map on µ m also lies in 2 Aut( µ m ) and has the same restriction to µ e as τ . Hence, τ | µ m m ust b e the in v erse map. That is, { ω ∈ µ n ( T 0 ) | τ ( ω ) = ω − 1 } = µ n ( T 0 ). Therefore, ( 5.5 ) ab o v e sh o ws that im( α ) = µ n ( T 0 ) /µ e , which we n oted ab o v e is isomorp hic to SK 1 ( E ). UNIT AR Y SK 1 OF DIVISION ALGEBRAS 23 (iii) W e saw in the p ro of of part (i) that τ acts on µ e b y the in v erse map. S o, if e > 2, then µ e 6⊆ R 0 . Since [ T 0 : R 0 ] = 2, it then follo ws that T 0 = R 0 ( µ e ).  R emark 5.2 . Th e isomorphism SK 1 ( E , τ ) ∼ = SK 1 ( E ) of part ( ii ) of the ab o ve theorem can b e obtained under the milder condition that E 0 = T 0 E ′ pro vided that the exp onent of E is a pr im e p ow er. Th e proof is similar. Example 5.3 . Let r 1 , . . . , r m b e in tegers with eac h r i ≥ 2. Let e = lcm( r 1 , . . . , r m ), and let n = r 1 . . . r m . Let C b e any ﬁ eld su c h that µ e ⊆ C and C has an automorphism θ of order 2 suc h that θ ( ω ) = ω − 1 for all ω ∈ µ e . Let R b e the ﬁxed ﬁeld C θ . Let x 1 , . . . , x 2 m b e 2 m indep en d en t indeterminates, and let K b e the iterated Lauren t p o wer series ﬁeld C (( x 1 )) . . . (( x 2 m )). This K is equip p ed with its standard v aluation v : K ∗ → Z 2 m where Z 2 m is giv en the righ t-to-left lexicog r aphical o r dering. With this v aluation K is henselian (see [ W 2 , p. 397]). Consider the tensor p ro duct of symbol algebras D =  x 1 , x 2 K  ω 1 ⊗ K . . . ⊗ K  x 2 m − 1 , x 2 m K  ω m , where for 1 ≤ i ≤ m , ω i is a primitive r i -th root of u nit y in C . Usin g the v aluation theory devel op ed for division algebras, it is kn o wn that D is a division algebra, the v aluation v e xtend s to D , and D is totally ramiﬁed o ver K (see [ W 2 , Ex. 4.4(ii)] and [ TW , Ex. 3.6]) w ith Γ D / Γ K ∼ = m Q i =1 ( Z /r i Z ) × ( Z /r i Z ) , and D = K ∼ = C . Extend θ to an automorphism θ ′ of order 2 on K in the ob vious w a y , i.e., act- ing b y θ on the co eﬃcient s of a Lauren t series, and with θ ′ ( x i ) = x i for 1 ≤ i ≤ 2 m . On eac h of the sym b ol algebras  x 2 i − 1 ,x 2 i K  ω i with its generators i i and j i suc h that i r i i = x 2 i − 1 , j r i i = x 2 i , and i i j i = ω i j i i i , deﬁ n e an in vo lution τ i as follo ws: τ i ( c i k i j l i ) = θ ′ ( c ) j l i i k i , where c ∈ K and 0 ≤ l, k < r i . Clearly K τ i = K θ ′ = R (( x 1 )) . . . (( x 2 m )), and therefore τ i is a unitary in v olution. Sin ce the τ i agree on K for 1 ≤ i ≤ m , they yield a un itary in vo lu tion τ = ⊗ m i =1 τ i on D . Now by Th. 3.5 , SK 1 ( D , τ ) ∼ = SK 1 ( gr ( D ) , e τ ). Since D is totally ramiﬁed o ve r K , whic h is unramiﬁed ov er K τ , w e hav e corresp ondingly that gr ( D ) is tot ally ramiﬁed o ve r gr ( K ), which is unramiﬁ ed o v er gr ( K ) e τ . Also, gr ( K ) 0 ∼ = K ∼ = C . W e ha ve exp( gr ( D )) = exp ( D ) = exp(Γ D / Γ K ) = lcm( r 1 , . . . , r m ) = e and ind( gr ( D )) = ind( D ) = r 1 . . . r m = n . By Th. 5.1 , SK 1 ( D , τ ) ∼ = SK 1 ( gr ( D ) , e τ ) ∼ = { ω ∈ µ n ( C ) | θ ( ω ) = ω − 1 }  µ e , while b y [ HaW , Th. 4.8, Cor. 3.6(ii)], SK 1 ( D ) ∼ = SK 1 ( gr ( D )) ∼ = µ n ( C ) /µ e . Here are some more sp eciﬁc examples: (i) Let C = C , the complex num b ers, and let θ b e complex conjugation, which maps every r o ot of unity to its inv ers e. So, R = C θ = R . Then , S K 1 ( D , τ ) ∼ = SK 1 ( D ) ∼ = µ n /µ e ∼ = Z / ( n/e ) Z . (ii) L et r 1 = r 2 = 4, so e = 4 and n = 16. Let ω 16 b e a primitiv e sixteen th ro ot of unit y in C , and let C = Q ( ω 16 ), the sixteenth cyclotomic extension of Q . Recall that Gal( C / Q ) ∼ = Aut( µ 16 ) ∼ = ( Z / 4 Z ) × ( Z / 2 Z ), Let θ : C → C b e the automorphism which maps ω 16 7→ ( ω 16 ) 7 . Then, θ 2 = id C , as 7 2 ≡ 1 (mo d 16), and { ω ∈ µ 16 | θ ( ω ) = ω − 1 } = µ 8 . Thus, SK 1 ( D , τ ) ∼ = µ 8 /µ 4 ∼ = Z / 2 Z , while S K 1 ( D ) ∼ = µ 16 /µ 4 ∼ = Z / 4 Z . So, here the injection S K 1 ( D , τ ) → SK 1 ( D ) is not surjective . (iii) Let r 1 = . . . = r m = 2, s o e = 2 and n = 2 m . Here, C could b e any quadr atic extension of an y ﬁeld R with c h ar( R ) 6 = 2. T ak e θ to b e the u nique nonidenti ty R -automorphism of C . The resulting D is a tensor pro du ct of m quaternion algebras o ver C (( x 1 )) . . . (( x 2 m )), and S K 1 ( D , τ ) ∼ = { ω ∈ µ 2 m ( C ) | θ ( ω ) = ω − 1 }  µ 2 , while SK 1 ( D ) ∼ = µ 2 m ( C ) /µ 2 . Ex. 5.3 giv es an ind ication ho w to use the graded approac h to reco ve r results in the literature on the unitary SK 1 in a u niﬁed manner and to extend them from division algebras with discrete v alued groups 24 R. H AZRA T AND A. R. W AD SW OR TH to arbitrary v alued group s. While SK 1 ( D ) has long b een kno wn for the D of Ex. 5.3 , the f orm ula for SK 1 ( D , τ ) is new. Here is a more complete statemen t of what th e results in the preceding s ections yield for SK 1 ( D , τ ) for v alued d ivision alg ebr as D . Theorem 5.4. L et ( D , v ) b e a tame value d division algebr a over a ﬁeld K with v | K henselian, with a unitary involution τ ; let F = K τ , a nd supp ose v | F is henselian and that K is tamely r amiﬁe d over F . L et τ b e the involution on D induc e d by τ . Then, (1) Supp ose K is unr amiﬁe d over F . (i) If D i s unr amiﬁe d over K , then S K 1 ( D , τ ) ∼ = SK 1 ( D , τ ) . (ii) If D is total ly r amiﬁe d over K , let e = exp( D ) and n = in d( D ) ; then, SK 1 ( D , τ ) ∼ = { ω ∈ µ n ( K ) | τ ( ω ) = ω − 1 }  µ e , while SK 1 ( D ) ∼ = µ n ( K ) /µ e . (iii) If D has a maximal gr ade d su b ﬁeld M unr amiﬁe d over K and another maximal gr ade d subﬁeld L tota l ly r amiﬁe d over K , with τ ( L ) = L , th en D is semir amiﬁe d and SK 1 ( D , τ ) =  a ∈ D ∗ | N D /K ( a ) ∈ F   Q h ∈ Gal( D /K ) F ∗ h τ . (iv) Supp ose Γ D / Γ K is c yclic. L et σ b e a gener ator of Gal( Z ( D ) /K ) . Then, SK 1 ( D , τ ) ∼ = { a ∈ D ∗ | N Z ( D ) /K (Nrd D ( a )) ∈ F }   Σ τ ( D ) · Σ σ τ ( D )  . (v) If D is inertial ly split, D is a ﬁeld and G al ( D /K ) is cyclic, then SK 1 ( D , τ ) = 1 . (2) If K is total ly r amiﬁe d over F , th en SK 1 ( D , τ ) = 1 . Pr o of. Let gr ( D ) b e the asso ciated graded division algebra of D . Th e ta men ess assu mptions assure that gr ( K ) is th e cente r of gr ( D ) with [ gr ( D ) : gr ( K )] = [ D : K ] and that th e graded inv olution e τ on gr ( D ) induced b y τ is u n itary with gr ( K ) e τ = gr ( K τ ). In eac h case of Th. 5.4 , the conditions on D yield analogous conditions on gr ( D ). Since by Th . 3.5 , S K 1 ( D , τ ) ∼ = SK 1 ( gr ( D ) , e τ ), (2) and (1)(v) follo w immediately from Prop. 4.5 and Prop. 4.13 ( v ), r esp ectiv ely . P art (1) (i), also follo ws fr om Th. 3.5 , and Cor. 4.10 as f ollo w s : SK 1 ( D , τ ) ∼ = SK 1 ( gr ( D ) , e τ ) ∼ = SK 1 ( gr ( D ) 0 , τ | gr ( D ) 0 ) = SK 1 ( D , τ ) . P arts (1)(ii), (1)(iii), and (1)(iv) follo w similarly using T h. 5.1 , Cor. 4.11 , and Prop. 4.13 ( ii ) resp ectiv ely .  In the sp ecial case that the henselian v aluation o n K is discrete (rank 1), Th. 5.4 (1)(i), (iii) , (iv), (v) and (2) we re obtained b y Y anc hevski ˘ ı [ Y 2 ]. 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Unitary SK1 of graded and valued division algebras, I

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