Quantum group actions on rings and equivariant K-theory

QUANTUM GR OUP A CTIONS ON RINGS AND EQUIV ARIANT K-THEOR Y G. I. LEHRER AND R. B. ZHANG Abstract. Let U q ( g ) b e a quantum group. Regarding a (noncommut ative) space with U q ( g )-symmetry as a U q ( g )-mo dule alg ebra A , we may think of equiv arian t vector bundles on A as pro jective A -mo dules with compatible U q ( g )-action. W e construct a n equiv ariant K-theor y of s uch qua nt um vector bundles us ing Quillen’s exact categ ories, and pr ovide mea ns for its compution. The equiv ariant K -groups of quantum homogeneous spac e s and quantum symmetric alge br as of classica l t yp e are computed. Contents 1. In tro duction 1 2. Equiv ariant K-theory o f q uan tum group actions 4 2.1. Mo dule algebras and equiv arian t mo dules 4 2.2. Equiv ariant K-theory of quan tum group actions 6 3. Categories of equiv ariant modules 7 4. Equiv ariant K-theory o f ﬁlte red mo dule algebras 10 4.1. Graded mo dule algebras 11 4.2. Filtered mo dule algebras 12 5. Quan tum symmetric a lgebras 16 5.1. Equiv ariant K-theory of quan tum sy mmetric algebras 16 5.2. Quadratic algebras and Koszul complexes 16 5.3. Pro of of Theorem 5.3 20 5.4. Examples 20 6. Quan tum homogeneous s paces 24 6.1. Quan tum homogenous space s 24 6.2. Equiv ariant K -theory of quan tum homogeneous spaces 25 6.3. Pro of of Theorem 6.4 27 App endix A. Como dules and smash pro ducts 29 References 30 1. Intr oduction In rece n t y ears there has b een m uc h w ork exploring v arious t yp es of noncomm u- tativ e geometries with p ossible applications in phy sics. The b est deve lop ed theory is Conne s’ noncomm utativ e diﬀeren tial geometry [10] form ulated within the frame- w ork of C ∗ -algebras, whic h incorp orates K-theory and cyclic cohomology and has 2000 Mathematics S u bje ct Classiﬁc ation. 19L47, 20G42, 17B37, 58B32 , 81R50 . 1 2 G. I. LEHRER AND R. B. ZHAN G yielded new index theorems. V arious asp ects of noncomm utativ e alg ebraic geometry ha v e also b een dev elop ed (see, e.g., [3, 29]). Noncomm utative generalisations of classical geometries are based o n the strategy of regarding a s pace as deﬁned b y an algebra of functions, whic h is commutativ e in the class ical case. In noncomm utative geometry [10] one replaces this commutativ e algebra b y noncomm utative a lgebra; in analog y with t he clas sical case, vector bun- dles a r e regarded as ﬁnitely generated pro jective mo dules o v er this algebra. One ma y then inv estigate problems with g eometric origins by means of these algebraic structures. This p ermits cross fertilisation o f algebraic and geometric ideas, and is exp ected t o lead to mathematical adv ances in b o th areas. An imp ortant motiv at ion from ph ysics f or studying noncomm utat ive geometry is the notion that spacetime at the Planc k scale b ecomes noncommutativ e [1 3 ]; thu s noncomm utativ e geometry ma y b e a necessary ingredien t for a consisten t t heory of quantum gra vit y . Muc h o f classical algebraic and diﬀeren tial geometry concerns algebraic v arieties and manifolds with a lgebraic or Lie group actions. Corresp ondingly , in no ncom- m utat iv e geometry one studies noncommu tativ e algebras with Hopf algebra ac- tions. Natural examples of suc h noncomm utativ e geometries are quan tum a na logues [15, 33, 34, 19] of homogeneous spaces and homogeneous v ector bundles [7], whic h ha v e prov ed useful for form ulating the quantum group vers ion [1] of the Bott-Borel- W eil theorem [7] in t o a noncomm ut a tiv e geometric setting [15, 33]. Some q uan tum homogeneous spaces suc h as quan tum spheres hav e b een particular ob jects of a t t en- tion [11, 20, 19] b ecause of p otential phy sical applications. W e note that inte resting examples of Hopf a lgebras acting on no ncomm utative algebras are no nco comm uta- tiv e, and th us do not correspo nd to g roups. In this paper we study noncommu tativ e geometries with quantum group symme- tries. In particular, w e shall study an equiv ariant algebraic K-t heory of suc h spaces whic h is a generalisation of the equiv a rian t a lg ebraic K-theory of reductiv e g roup actions inv estigated b y Bass and Hab oush in [4, 5]. The equiv aria nt top olog ical K-theory of Lie group actions and algebraic group sc heme actions ha v e b een dev el- op ed in the celebrated pap ers [28 , 12, 30]. F ollo wing the w ork of Bass and Hab oush, sev eral authors ha v e addressed geometric themes in t he con text of the equiv arian t K-theory of algebraic v ector bundles [6, 21, 22]. A recen t t reatmen t of the K- theory of compact Lie group actio ns in relation to represen tation theory ma y b e fo und in [17, § 12]. The need for a n equiv a r ian t K-theory in the noncomm utativ e setting was already clear in the classiﬁcation of quan tum homogeneous bundles [34]. V ery recen tly , Nest and V oigt ha v e extende d the notion o f P oincare duality in K- t heory to the setting of compact q uan tum group actions [24] within the fra mework of C ∗ -algebras. In our situation, the t w o crucial notions whic h are needed are those of a n ‘equi- v ar ia n t noncommutativ e space’, whic h w e shall tak e to mean a mo dule algebra A o v er a Hopf algebra [23], and an ‘equiv arian t vec tor bundle on A ’ whic h we shall tak e to mean an equiv aria nt mo dule o v er the mo dule a lgebra A . Sp eciﬁcally , a mo dule algebra is a n asso ciativ e a lgebra that is a lso a mo dule for the Hopf algebra, whose algebraic structure is pres erv ed by the action; equiv ariant mo dules o ver module al- gebras w ere in tro duced in [34] in the con text of quan tum homogeneous s paces, and w e generalise it he re to arbitrary mo dule algebras. EQUIV ARIANT K-THEOR Y OF QUANTUM GROUP A CTIONS 3 Let U = U q ( g ) b e a quan tum group deﬁned ov er the ﬁeld k = C ( q ), and let A b e a mo dule algebra ov er U. W e in tro duce the category M ( A, U) of U-equiv ariant A -mo dules whic h are ﬁnitely A -generated and lo cally U-ﬁnite. The full sub category P ( A, U) of M ( A, U) consisting of ﬁnitely A -generated, lo cally U-ﬁnite, pro jectiv e equiv ariant mo dules is an exact c ategory in a natural wa y . Thus Quillen’s K-theory of exact categories applies to P ( A, U), giving rise to an algebraic K-theory of mo dule algebras whic h is equiv arian t under the action of the q uan tum group U (Deﬁnition 2.4). Prop erties of equiv ariant K- theory a re dev elop ed for mo dule algebras w ith ﬁltra- tions whic h are stable under quantum group actions. Under a regularity assumption for the mo dule algebras, w e establish a relationship b etw een the K-groups of the ﬁl- tered algebras and the degree zero subalgebras of the corresp onding graded algebras (see Theorem 4 .3 for details). This may b e regarded as an equiv ariant analogue of [27, § 6, Theorem 7] on the higher algebraic K-theory of ﬁltered rings. W e apply Theorem 4.3 to compute the equiv arian t K-g roups for a class of mo dule algebras ov er quan tum groups, whic h we call quan tum symmetric alg ebras. The results are summarised in Theorem 5.3. In establishing the regularity of the left No etherian quantum symmetric algebras, some elemen ts of the theory of Koszul algebras [26] are used, whic h are dis cussed in Section 5.2 . One of the main motiv a tions of [5] is to pro v e the results [5 , Theorems 1.1,2.3 , Cor. 2.4], whic h w ould be easy conse quences of the Serre conj ecture if one assumed that all reductiv e group actions on a ﬃne space are linearisable. Th us a natural question whic h arises f r o m o ur w ork is whe ther there is a non-comm utativ e v ersion of the Se rre c onjecture f or the quantum symmetric algebras we consider. In [2], the case of the natural repres en ta tion of U q ( gl n ) is discussed. W e also study the equiv aria n t K-theory of quan tum homog eneous spaces in detail. Giv en a quantum homogeneous space o f a quantum group U q ( g ) whic h corresp onds to a reductiv e quantum subgroup U q ( l ), we show that the equiv ariant K-groups are isomorphic to the K - groups of the exact category whose ob jects ar e the U q ( l )- mo dules (see Corolla ry 6.5 for the precise statement). Prop erties of the categories M ( A, U) a nd P ( A, U ), analogous to those of their classical ana lo gues are established, and used in the study of equiv ariant K-theory . W e pro v e a splitting lemma (Prop osition 3.2), whic h enables us to c haracterise the ﬁnitely A generated, lo cally U-ﬁnite, pro jectiv e U-equiv ariant A -mo dules (Corollary 3.4). A similar result in the comm uta t iv e setting w as prov ed b y Bass and Hab oush [4, 5] for reductiv e algebraic group actions. The equiv arian t algebraic K-t heory constructed here generalises, in a completely straigh tforw ard manner, to Hopf algebras whose lo cally ﬁnite mo dules a r e semi- simple, e.g., univ ersal env eloping algebras o f ﬁnite dimensional semi-simple Lie al- gebras. In f a ct, the case of such univ ersal en v eloping algebras essen tially co v ers the Bass-Hab oush theory fo r semi-simple algebraic gro up actions when the mo dule alge- bras a re commutativ e. W e also p o int out that t he equiv arian t alg ebraic K-theory of a U-mo dule algebra A de v elop ed here is diﬀeren t from the usual algebraic K- theory of the smash pro duct algebra R := A #U, see Remark 2.6. The organisation of the pap er is as follows. In Section 2, w e in tr o duce v arious categories of equiv ariant mo dules for mo dule algebras ov er quan t um groups, and deﬁne the equiv arian t a lgebraic K-theory of quan tum group actions. In Section 3, 4 G. I. LEHRER AND R. B. ZHAN G the theory of equiv ariant mo dules is deve lop ed, a nd is used to study quantum gro up equiv ariant K-theory . In Section 4 w e dev elop the equiv ariant K-theory of ﬁltered mo dule algebras, and in the remaining tw o sec tions w e study concrete examples. In Section 5, w e compute the equiv aria nt K- groups of the quantum symmetric alge- bras, a nd in Sec tion 6 w e inv estigate in detail the equiv arian t K-groups of quantum homogeneous spaces. 2. Equiv ariant K-the or y of quantum group actions The purp ose o f this section is to in tro duce an equiv arian t algebraic K-theory of quantum gr oup actions. This theory generalises, in a straig h tf o rw ard wa y to arbitrary Hopf alg ebras. 2.1. Mo dule algebras and equiv arian t mo dules. F or any ﬁnite dimensional simple complex Lie algebra g , denote b y U := U q ( g ) the quan tum gro up deﬁned o v er the ﬁeld k = C ( q ) of ratio nal functions in q ; U has a standard presen tatio n with generators { e i , f i , k ± 1 i | i = 1 , . . . , r } ( r = ra nk( g )), and relations whic h may b e found e.g., in [1]. If g is a semi-simple Lie algebra, U will denote the tensor pro duct of the quan tum groups in t he ab ov e sen se, asso ciated with t he sim ple factors. It is w ell kno wn that U has the structure of a Hopf algebra; denote its co- m ultiplication b y ∆, co-unit b y ǫ and antipo de by S . W e shall use Sw eedler’s notation for co-m ultiplication: giv en an y x ∈ U, write ∆( x ) = P ( x ) x (1) ⊗ x (2) . The follo wing relations a re among those satisﬁed b y an y Hopf algebra: X ( x ) ǫ ( x (1) ) x (2) = X ( x ) x (1) ǫ ( x (2) ) = x, X ( x ) S ( x (1) ) x (2) = X ( x ) x (1) S ( x (2) ) = ǫ ( x ) . Let ∆ ′ b e the opp o site co-m ultiplication, deﬁned by ∆ ′ ( x ) = P ( x ) x (2) ⊗ x (1) for any x ∈ U. Denote by U- mo d the category of ﬁnite dimensional left U-mo dules of t yp e- (1 , . . . , 1). Then U- mo d is a semi-simple braided tensor category . A (left) U- mo dule V is called lo c al ly ﬁnite if for a n y v ∈ V , the cyclic submo dule U v g enerated b y v is ﬁnite dimensional. W e shall mak e use of the imp ortant fact tha t lo cally ﬁnite mo dules are semi-simple. W e shall sa y that a lo cally ﬁnite U-mo dule is ty p e- (1 , . . . , 1) if all its ﬁnite dimensional submo dules are t yp e-(1 , . . . , 1). An asso ciative algebra A with iden tit y 1 is a (left) mo dule algebr a over U [23] if it is a left U-mo dule, and the mu ltiplication A ⊗ k A − → A and unit map k − → A are U- mo dule homomorphisms. Explicitly , if w e write the U-a ctio n o n A as U ⊗ k A − → A , x ⊗ a 7→ x · a , for all a ∈ A and x ∈ U, then x · ( ab ) = X ( x ) ( x (1) · a )( x (2) · b ) , x · 1 = ǫ ( x )1 . W e call a U- mo dule algebra A lo c a l ly ﬁnite if it is lo cally ﬁnite as a U- mo dule. If all its submodules are in U- mo d , w e sa y tha t the locally ﬁnite U- mo dule algebra is t yp e-(1 , . . . , 1). EQUIV ARIANT K-THEOR Y OF QUANTUM GROUP A CTIONS 5 An elem en t a ∈ A is U -inv a riant if x · a = ǫ ( x ) a for all x ∈ U. W e de note b y A U the subm o dule of U-in v ariants of A , that is, A U := { a ∈ A | x · a = ǫ ( x ) a, ∀ x ∈ U } . The fact that U is a Hopf algebra implies that this is a subalgebra of A . Indeed, for all a, b ∈ A U , w e ha v e x · ( ab ) = X ( x ) ( x (1) · a )( x (2) · b ) = X ( x ) ǫ ( x (1) ) ǫ ( x (2) ) ab = ǫ ( x ) ab. Hence ab ∈ A U . W e shall refer to A U as the sub algeb r a o f U -i n variants of A . Let M b e a left A -mo dule with structure map φ : A ⊗ M − → M , and also a lo cally ﬁnite left U-mo dule with structure map µ : U ⊗ M − → M . Then A ⊗ k M has a natural U-mo dule structure µ ′ : U ⊗ ( A ⊗ M ) − → A ⊗ M , x ⊗ ( a ⊗ m ) 7→ X ( x ) x (1) · a ⊗ x (2) m. The A -mo dule and U-mo dule structures of M are said to b e compatible if t he follo wing diagram commutes (2.1) U ⊗ ( A ⊗ M ) id ⊗ φ − → U ⊗ M µ ′ ↓ µ ↓ A ⊗ M φ − → M . In this case, M is called a U -e quivaria nt left A -mo dule , or A -U-mo dule for simplicit y . A morphism b etw een tw o A -U-mo dules is an A -mo dule map whic h is at the same time also U-linear. W e denote b y Hom A -U ( M , N ) the space of A -U-morphisms. Denote b y A - U- mo d the category of lo cally U- ﬁnite A - U-mo dules (i.e., lo cally U- ﬁnite U-equiv ariant left A -mo dules), which as U-mo dules a re of ty p e-(1 , . . . , 1 ). It is clear tha t A -U- mo d is an ab elian category . Let M ( A, U) b e the f ull sub cat ego ry of A -U-mo dules consisting of ﬁnitely A -generated ob jects, and de note b y P ( A, U) the full subcategory of M ( A, U) whose ob jects are the pro jective ob jects in A -U- mo d . Remark 2.1. (1) In this w ork, U shall generally denote U = U q ( g ), where g is a reductiv e Lie alg ebra. In p erticular, It may ha pp en that the ro ot lattice has smaller ra nk t han the we igh t lat t ice. W e shall consider lo cally ﬁnite U-mo dules and lo cally ﬁnite U-mo dule algebras whic h are of t yp e-(1 , . . . , 1) for the c hosen dominan t w eights . In pa r ticular, the categories of ﬁnite di- mensional mo dules considered are semisimp le. (2) The c ategories just intro duced are q uan tum analogues o f those o ccurring in [5, T heorem 2.3]. W e also deﬁne a U -e quiva riant right A -mo dule M , as a left U-mo dule whic h is also a righ t A -module, suc h that the mo dule structures are compatible in t he sense that x ( ma ) = X ( x ) ( x (1) m )( x (2) · a ) 6 G. I. LEHRER AND R. B. ZHAN G for all x ∈ U, a ∈ A a nd m ∈ M . Similarly , w e also deﬁne a U -e quiva ri a nt A - bimo dule M , as a left U-mo dule whic h is also an A -bimo dule, suc h that x ( amb ) = X ( x ) ( x (1) · a )( x (2) m ( x (3) ) · b ) for all x ∈ U, a, b ∈ A and m ∈ M . Let R b e a U-equiv a rian t righ t A -mo dule, and let B b e a U-equiv ariant A - bimo dule. F or an y U-equiv ariant left A -mo dule M , R ⊗ A M has the structure of left U-mo dule, and B ⊗ A M has the structure of U-equiv a rian t left A -mo dule, with the mo dule s tructures deﬁne in the follow ing w a y . F or all r ∈ R , b ∈ B , a ∈ A and m ∈ M , x ( r ⊗ A m ) = X ( x ) x (1) r ⊗ A x (1) m, x ( b ⊗ A m ) = X ( x ) x (1) b ⊗ A x (1) m, a ( b ⊗ A m ) = ab ⊗ A m. The follo wing result is now clear. Lemma 2.2. L et A b e a lo c al ly ﬁnite U -mo dule a l g ebr a. L et R b e a U -e quiva riant right A -mo dule, and le t B b e a U -e quivarian t A -bimo dule. Assume that b oth R and B ar e lo c al ly U -ﬁnite, then we have c ovarian t functors R ⊗ A − : A - U - mo d − → U - Mo d l.f . , B ⊗ A − : A - U - mo d − → A - U - mo d , wher e U - Mo d l.f . is the c ate gory of lo c al ly ﬁnite U -mo dules. F urthermor e, if B is also ﬁnitely A -gene r a te d a s a l e f t A -mo d ule, then we have the c ova ri a nt functor B ⊗ A − : M ( A, U) − → M ( A, U) . In this w ork, the term ‘mo dule’ w ill mean left mo dule unless otherwis e stated. 2.2. Equiv arian t K-theory of quantum group actions. Recall that an exact category P is an additiv e category with a class E of short exact sequences which satisﬁes a series o f a xioms, see [27, p.99 ] or [16, App endix A]. F or our purp oses, w e may think of an exact catego ry P as a full (additiv e) sub category of an a b elian category A whic h is closed under extensions in A , that is, for a n y short exact sequence 0 − → M ′ − → M − → M ′′ − → 0 in A , if M ′ and M ′′ are in P , then M also b elongs to P . T ypical examples of exact categories are (1) an y a b elian category with exact structure giv en b y all short exact sequenc es, and (2) the full sub category of ﬁnitely g enerated pro jects (left) mo dules of t he cat- egory of (left) mo dules o ve r a ring. F or any exact category P in whic h the isomorphism classes o f ob jects form a set, one may deﬁne the Quillen category QP . Quillen’s algebraic K-gro ups [27] of the exact category P ar e deﬁned to b e t he homotop y groups of the classifying space B ( QP ) of QP : K i ( P ) = π i +1 ( B ( QP )) , i = 0 , 1 , . . . . EQUIV ARIANT K-THEOR Y OF QUANTUM GROUP A CTIONS 7 If F : P 1 − → P 2 is an exact functor b et we en exact categories, it induces a functor Q F : QP 1 − → QP 2 b et w een the corresp onding Quillen categories. This functor then induces a cellular map B Q F : B ( QP 1 ) − → B ( QP 2 ), whic h in turn leads to the homomorphisms F ∗ : K i ( P 1 ) − → K i ( P 2 ) , for all i. W e no w turn to the deﬁnition of an equiv arian t algebraic K-theory of quan tum group actions. The follo wing fact is immediate from t he deﬁ nition of P ( A, U). Theorem 2.3. L e t A b e a lo c al ly ﬁnite mo dule algebr a over the quantum gr oup U . Th en the c ate g o ry P ( A, U) of ﬁnitely A -gen er ate d, lo c al ly U -ﬁnite, pr oj e ctive U -e quivariant A -mo dules is an exact c ate gory. The Quillen K-groups K i ( P ( A, U)) are therefore deﬁned for P ( A, U), a nd the follo wing deﬁnition mak es sense . Deﬁnition 2.4. L et A b e a lo c al ly ﬁnite mo d ule algebr a ov e r the quantum gr oup U . The U -e quivariant al g ebr aic K-gr oups of A ar e deﬁne d by K U i ( A ) := K i ( P ( A, U)) , i = 0 , 1 , . . . . It f ollo ws from standard facts [27, T heorem 1, p.102] tha t the fundamen tal gro up of B ( QP ( A, U)) is isomorphic to the Grothendiec k group of P ( A, U). Hence the U-equiv ariant K-group K U 0 ( A ) is isomorphic to the Gro thendiec k group of P ( A, U). Remark 2.5. The U-equiv arian t algebraic K-g roups K U i ( A ) of A are a generalisa- tion to the quan tum group setting of the equiv a r ia n t algebraic K- groups of reductiv e algebraic group actions studied in [4, 5]. Remark 2.6. One may also consider the usual algebraic K-theory of the smash pro duct A # U (see App endix A for the deﬁnition of a smash pro duct). This, ho wev er, is completely diﬀeren t f r om the equiv aria n t K-theory of the U-mo dule alg ebra A in tr o duced here. See App endix A fo r more details. 3. Ca tegories of equiv ariant modules T o study the equiv arian t K- groups intro duced in the last section, w e require some prop erties of v arious categories of equiv ariant mo dules. Fix a quan tum group U and a lo cally ﬁnite mo dule alg ebra A ov er U. As w e hav e already declared in Remark 2.1, all lo cally ﬁnite U-mo dules and lo cally ﬁnite U-mo dule algebras considered are assumed to b e t yp e-(1 , . . . , 1 ). Lemma 3.1. L et M a n d N b e A - U -mo d ules . Then ther e is a natur al U -a ction on Hom A ( M , N ) de ﬁne d for any x ∈ U an d f ∈ Hom A ( M , N ) by ( xf )( m ) = X ( x ) x (2) f ( S − 1 ( x (1) ) m ) , ∀ m ∈ M . (3.1) 8 G. I. LEHRER AND R. B. ZHAN G Pr o of. W e ﬁrst sho w that (3.1) deﬁnes a U-mo dule structure on Hom k ( M , N ). F or an y x, y ∈ U and f ∈ Hom k ( M , N ), w e ha v e ( y ( xf ))( m ) = X ( y ) y (2) ( xf )( S − 1 ( y (1) ) m ) = X ( y ) , ( x ) y (2) x (2) f ( S − 1 ( x (1) ) S − 1 ( y (1) ) m ) for all m ∈ M . Using the facts that for all x, y ∈ U, S ( y x ) = S ( x ) S ( y ) and ∆( y x ) = P ( x ) , ( y ) y (2) x (2) ⊗ y (1) x (1) , we c an cast the far righ t hand s ide in to X ( yx ) ( y x ) (2) f ( S − 1 (( y x ) (1) ) m ) = (( y x ) f )( m ) Th us Hom k ( M , N ) is a U-mo dule. Next w e sho w that Hom A ( M , N ) is a U- submo dule o f Hom k ( M , N ). Let f ∈ Hom A ( M , N ), a ∈ A and x ∈ U. Then for all m ∈ M , w e ha ve ( xf )( am ) = X ( x ) x (2) f ( S − 1 ( x (1) )( am )) = X ( x ) x (3) f (( S − 1 ( x (2) ) · a ) S − 1 ( x (1) ) m ) = X ( x ) x (3) (( S − 1 ( x (2) ) · a ) f ( S − 1 ( x (1) ) m )) , where the last step used the fact that f is A -linear. The far right hand side can b e rewritten as P ( x ) ( x (3) · ( S − 1 ( x (2) ) · a )) x (4) f ( S − 1 ( x (1) ) m ). By using the deﬁning prop ert y of the antipo de, we can further simplify it to obtain X ( x ) ǫ ( x (2) ) a  x (3) f ( S − 1 ( x (1) ) m )  = X ( x ) a  x (2) f ( S − 1 ( x (1) ) m )  = a ( xf )( m ) . Hence xf ∈ Hom A ( M , N ), as required.  F or any M , N in M ( A, U), the U-action on Hom A ( M , N ) deﬁned in Lemma 3.1 is semi-simple, and Hom A ( M , N ) b elongs to M ( A, U). T o see this, w e note t hat there exists a ﬁnite dimensional U-mo dule V whic h generates M ov er A . Th us Hom A ( M , N ) is isomorphic to a submo dule of V ∗ ⊗ k N as a U-mo dule, which is ob viously lo cally ﬁnite and th us semi-simple o v er U. Prop osition 3.2. (Splitting lemma) Co n sider a short exac t se quenc e 0 − → M ′ − → M p − → M ′′ − → 0 (3.2) in A - U - mo d , wher e M ′′ is an obje c t of M ( A, U ) . I f the exac t se quenc e is A -split, then it is also split as an exa c t se quenc e of A - U -mo dules. EQUIV ARIANT K-THEOR Y OF QUANTUM GROUP A CTIONS 9 Pr o of. Since the ab o v e sequence is A -split, there is an A -mo dule isomorphism M ∼ − → M ′ ⊕ M ′′ . Therefore Hom A ( M ′′ , M ) p ◦− − → Hom A ( M ′′ , M ′′ ) − → 0 is exact. This is an exact sequence of U-mo dules a s the map p ◦ − is clearly U-linear. Since M ′′ is an o b- ject of M ( A, U ) and M is lo cally U- ﬁnite, b oth hom-spaces are semi-simple U- mo dules. Hence w e hav e the exact sequence of U-in v a r ia n ts Hom A ( M ′′ , M ) U p ◦− − → Hom A ( M ′′ , M ′′ ) U − → 0. No te that fo r an y A - U -mo dules N and N ′ , an elemen t f of Hom A ( N , N ′ ) b elongs to Hom A -U ( N , N ′ ) if and only if xf = ǫ ( x ) f for all x ∈ U. Therefore, w e ha v e the exact sequence Hom A -U ( M ′′ , M ) p ◦− − → Hom A -U ( M ′′ , M ′′ ) − → 0 . No w an y elemen t of the pre-image of id M ′′ splits the exact sequence (3.2) of A - U- mo dules.  Giv en a ﬁnite dimensional U-mo dule V , w e deﬁne the free A -mo dule V A = A ⊗ k V with the ob vious A -action (giv en by left multiplication). W e also deﬁne a U-action on it by x ( a ⊗ v ) = X ( x ) x (1) · a ⊗ x (2) v for all a ∈ A , v ∈ V and x ∈ U. The se tw o actions a re easily b e show n to b e compatible. Call V A a fr e e A -U-mo dule of ﬁnite rank. Since the mo dule algebra A is locally U-ﬁnite, V A is also lo cally U-ﬁnite, and hence b elongs to M ( A, U). Lemma 3.3. F or e ac h obje ct M of M ( A, U) , ther e exists an e x a ct se quenc e V A − → M − → 0 in M ( A, U) , wher e V A is a fr e e A - U -mo dule of ﬁn ite r an k. Pr o of. G iv en an ob ject M o f M ( A, U), w e may c ho ose an y ﬁnite set of generato r s for it as a n A -mo dule, and consider the U-mo dule V generated b y this set. Then V is ﬁnite dimensional b ecause of the lo cal U-ﬁniteness o f M . W e hav e the obv ious surjectiv e A -U-mo dule map V A − → M , a ⊗ v 7→ av . Since A is lo cally U-ﬁnite, a free A -U-mo dule of ﬁnite rank is lo cally U-ﬁnite, th us the exact sequence V A − → M − → 0 is in M ( A, U).  As a corollary of P rop osition 3.2, w e ha v e the follo wing resu lt. Corollary 3.4. L et A b e a lo c al ly ﬁ n ite U -mo dule algebr a. F or any obje c t P of M ( A, U) , the fol low ing c onditions ar e e quivalen t : (1) P is pr oje ctive as an A -mo dule; (2) P is a pr oje ctive obje ct of A - U - mo d ; (3) P is a dir e ct summ and of some fr e e A - U -m o dule V A = A ⊗ k V of ﬁnite r ank. Pr o of. Assume that P is A -pro j ectiv e. Then give n an y exact sequence M → N → 0 in A -U- mo d , w e ha v e the e xact sequenc e Hom A ( P , M ) − → Hom A ( P , N ) − → 0 of U-mo dules. S ince b ot h Hom A ( P , M ) and Hom A ( P , N ) are semi-simple as U- mo dules, this le ads to the following exact sequenc e of k -v ector spaces Hom A − U ( P , M ) − → Hom A − U ( P , N ) − → 0 . 10 G. I. LEHRER AND R. B. ZHAN G This pro v es that (1) implies (2). No w assume that (2) holds. By L emma 3.3, there exists a free A - U-mo dule V A = A ⊗ k V of ﬁnite rank and an e xact sequence V A p − → P − → 0 o f A -U-modules. It follows from (2) that the identit y map of P factor s through p . Hence t he exact sequence splits in M ( A, U). This pro v es that (2 ) implies ( 3 ). It is eviden t t hat (3) implies (1), and t he prop osition follo ws.  Recall that an algebra A is called left r e gular if it is left No etherian and ev ery ﬁnitely gene rated left A -mo dule has a ﬁnite resolution by ﬁnitely generated pro jec- tiv e left A -mo dules. Prop osition 3.5. L et A b e a lo c al ly ﬁnite mo d ule algebr a over U . Assume that A is left r e gular. Th en every obje ct M in M ( A, U) admits a ﬁnite P ( A, U) -r esolution. Pr o of. L et M b e an ob ject in M ( A, U). By Lemma 3.3, there exists an exact sequence V 0 ,A p 0 − → M − → 0 in M ( A, U), where V 0 ,A = A ⊗ V 0 is a free A -U- mo dule of ﬁnite rank. Since K er ( p 0 ) also b elongs to M ( A, U) b ecause A is left No etherian, w e ma y apply the same considerations to it , a nd inductiv ely w e obtain an A - free resolution . . . − → V 1 ,A − → V 0 ,A − → M − → 0 in M ( A, U) for M . Let d b e the pro jectiv e dimension of M , which is ﬁnite b ecause A is regular. It follows from standard facts in homolog ical alg ebra (see, e.g., [32, Lemma 4.1.6]) that the k ernel P of the map V d − 1 ,A − → V d − 2 ,A is A -pro jectiv e, hence b elongs to P ( A, U) by Corollary 3.4. Th us w e arriv e at the P ( A, U)-resolution 0 − → P − → V d − 1 ,A − → . . . − → V 1 ,A − → V 0 ,A − → M − → 0 . This completes the pro of of t he proposition.  F or left regular mo dule algebra, we ha ve the following result, whic h is an analogue of [5, Theorem 2.3]. Prop osition 3.6. Assume that the lo c al ly ﬁnite U -mo dule algebr a A is left r e gular. Then ther e exis t the isomo rphisms K U i ( A ) ∼ − → K i ( M ( A, U)) , i = 0 , 1 , 2 , . . . . Pr o of. Since A is left regular, it m ust b e left No etherian. Th us M ( A, U) is an ab elian category , whic h has the nat ur a l exact structure consisting of all the short exact sequenc es. In view of Prop osition 3.5, t he em b edding P ( A, U) ⊂ M ( A, U) satisﬁes t he conditions o f Quillen’s Resolution Theorem [3 1, Theorem 4.6]. The statemen t follows.  4. Equiv ariant K-theor y of fil tered module algeb ras In this section w e dev elop prop erties of the equiv arian t K - theory of ﬁltered mo dule algebras o v er quan tum gr o ups. The main results here are Theorem 4.2 and Theorem 4.3, whic h are quantum ana logues of [5, T heorem 3.2, Theorem 4.1]. The pro o fs of these theorems a re ada pted from [5 , § 3, § 4] a nd [27, § 6]. EQUIV ARIANT K-THEOR Y OF QUANTUM GROUP A CTIONS 11 4.1. Graded mo dule algebras. Let S = ⊕ ∞ n =0 S n b e a graded, lo cally ﬁnite U- mo dule algebra. W e assume that the U-action preserv es the gr a ding o f S , t ha t is, eac h S n is stable under the U-action. Then A := S 0 , is a subalgebra of S . Set S + = ⊕ n> 0 S n . Then A ma y b e iden tiﬁed with S/S + . W e shall consider p ositiv ely graded U- equiv ariant S - mo dules, in whic h the U-action preserv es the gra ding. W e con tinue to assume that all mo dules are lo cally U-ﬁnite. F or (such) a graded S -U-mo dule N = ⊕ ∞ i =0 N i , we s et T i ( N ) = T or S i ( A, N ) , i ≥ 0 . These spaces hav e a natura l S -U-mo dule structure, which may b e seen as fol- lo ws. T ak e a sequence of graded U-equiv ariant S -mo dules whic h form a graded S -pro jectiv e resolution f or N ; this may b e done by taking, e.g., the usual normalised bar resolution for N , whic h is a sequence of graded U-equiv arian t S -mo dules as can b e easily seen by examining the explicit deﬁnition of the diﬀeren tial. By Lemma 2.2, the T or-groups T i ( N ) computed from such a resolution are S -U-mo dules. In particular, w e ha v e T 0 ( N ) = N /S + N with the natural S - U-mo dule structure . Let us introduce an increasing ﬁltration 0 = F − 1 N ⊂ F 0 N ⊂ F 1 N ⊂ . . . of N b y graded submo dules, b y taking F p N = P i ≤ p S N i . Then T 0 ( F p N ) n = 0 if n > p and T 0 ( F p N ) n = T 0 ( N ) n if n ≤ p . There is a lso a natural S -U-mo dule surjection S ( − p ) ⊗ A T 0 ( N ) p − → F p N/ F p − 1 N , (4.1) where S ( − p ) is S with the gra ding shifted b y p , that is, S ( − p ) n = S n − p . Remark 4.1. By [27, Lemma 1, p.117], if T 1 ( N ) = 0 and T or A i ( S, T 0 ( N )) = 0 for all i > 0, then the maps (4.1) are isomorphisms. Let M g r ( S, U) b e the additiv e category of ﬁnitely S -generated, p ositiv ely gra ded, and lo cally U-ﬁnite S -U-mo dules. If w e assume that S is left No etherian, then M g r ( S, U) is ab elian, and hence is an exact category . Its K -g r oups are naturally Z [ t ]-mo dules with t acting as the tra nslatio n functor N → N ( − 1). If S is A -ﬂat, then ev ery A -pro jectiv e resolution P • → V o f V in M ( A, U) giv es rise to an S -pro jectiv e resolution S ⊗ A P • → S ⊗ A V . Hence w e ha ve an exact functor ( S ⊗ A − ) : M ( A, U) → M g r ( S, U), whic h induces homomorphism s ( S ⊗ A − ) ∗ : K i ( M ( A, U)) − → K i ( M g r ( S, U)) (4.2) of K -groups. Theorem 4.2. Assume that S is left No etherian and A -ﬂat. I f A = S/S + has ﬁnite pr oje ctive dimension as an S -mo dule, then (4.2) extends to a Z [ t ] -mo dule isomorphism Z [ t ] ⊗ Z K i ( M ( A, U)) − → K i ( M g r ( S, U)) , for e ach i. (4.3) Pr o of. W e adapt the pro ofs of [2 7, Theorem 6] and [5, Theorem 3.2] to the presen t setting. Let M p denote the full sub category of M g r ( S, U) with ob jects N suc h that T i ( N ) = 0 for all i > p . If the pro jectiv e dimension of the S -mo dule A is d , then M 0 ⊂ M 1 ⊂ · · · ⊂ M d = M g r ( S, U). F or N in M p , Lemma 3.3 give s a surjectiv e S - U-map V S = S ⊗ k V ։ N , s ⊗ v 7→ sv , where V is a ﬁnite dimensional U-submo dule of N whic h generates N iself ov er S . Then the k ernel N ′ of the surjection b elongs to M g r ( S, U) since S is left No etherian. Because V S is a free S -mo dule, T i ( V S ) = 0 12 G. I. LEHRER AND R. B. ZHAN G for all i > 0. Hence the long exact sequence of T or groups arising from the short exact seque nce 0 → N ′ → V S → N → 0 yields T i ( N ) ∼ = T i − 1 ( N ′ ) for all i > 0 . This then implie s that N ′ b elongs to M p − 1 . Therefore, the inclusion M p ⊂ M p +1 for ev ery p ≥ 0 satisﬁes the conditions of the Resolution Theorem [31, Theorem 4.6], hence is a homotop y equiv alence. This leads to the homotop y equiv alence M 0 ⊂ M d = M g r ( S, U), whic h induces the isomorphisms K i ( M 0 ) ∼ − → K i ( M g r ( S, U)) , for all i = 0 , 1 , . . . . (4.4) Let V b e an ob ject in M ( A, U), and let P • → V b e an A - pro jectiv e resolution. Since S is A -ﬂat , S ⊗ A P • → S ⊗ A V is an S -pro jective resolution, and hence T i ( S ⊗ A V ) = 0 , f o r all i > 0 . Th us for any V in M ( A, U), S ⊗ A V b elongs to M 0 . Therefore, if M 0 ,n b e the full sub cat ego ry of M 0 whose ob j ects are mo dules M suc h that M = F n M , then w e ha v e an exact functor b : M ( A, U) n +1 − → M 0 ,n , ( V 0 , V 1 , . . . , V n ) 7→ ⊕ n p =0 S ( − p ) ⊗ A V p . This induces homomorphisms b ∗ : K i ( M ( A, U)) n +1 − → K i ( M 0 ,n ) . Since T 0 is ex act on M 0 , w e also hav e an ex act functor c : M 0 ,n − → M ( A, U) n +1 , M 7→ ( T 0 ( M ) 0 , T 0 ( M ) 1 , . . . , T 0 ( M ) n ) , and homomorphisms c ∗ : K i ( M 0 ,n ) − → K i ( M ( A, U)) n +1 . Note that c ◦ b is equiv alen t to the ide n tit y functor, th us c ∗ ◦ b ∗ = id. On t he other hand, an y M in M 0 ,n has a ﬁltra tion 0 = F − 1 M ⊂ F 0 M ⊂ F 1 M ⊂ · · · ⊂ F n M = M . Because of t he A -ﬂat nature of S , Remark 4.1 applies and we ha v e F p M F p − 1 M = S ( − p ) ⊗ A T 0 ( M ) p . Clearly eac h functor F p F p − 1 is exact. It fo llo ws from the addi- tivit y of c haracteristic ﬁltr a tions [27, Corollary 2, p.107] [3 1, Corollary 4.4] that P n p =0  F p F p − 1  ∗ = ( F n ) ∗ = id. No w observ e t ha t ( b ◦ c ) ∗ = P n p =0  F p F p − 1  ∗ , hence b ∗ ◦ c ∗ = id. P assing to the limit n → ∞ w e ha ve the follow ing isomorphism for eac h i : Z [ t ] ⊗ Z K i ( M ( A, U)) − → K i ( M 0 ) . Using the isomorphisms (4.4), w e arriv e at the desired result.  4.2. Filtered mo dule algebras. Let S b e a lo cally ﬁnite U- mo dule algebra. As- sume that S has an ascending ﬁltration 0 ⊂ F − 1 S ⊂ F 0 S ⊂ F 1 S ⊂ . . . suc h that 1 ∈ F 0 S , ∪ p F p S = S and F p S F q S ⊂ F p + q S . W e assume that the ﬁltration is preserv ed b y the U-action. Let S = g r ( S ) := ⊕ p ≥ 0 S p with S p := F p S/F p − 1 S, and set A = F 0 S and S + = ⊕ p> 0 S p . EQUIV ARIANT K-THEOR Y OF QUANTUM GROUP A CTIONS 13 Theorem 4.3. Assume that S is left No etherian and A -ﬂat. I f A ( = S /S + ) h as a ﬁnite pr oje ctive S -r esolution, then for i = 0 , 1 , 2 , . . . ther e exist i s omorphisms K i ( M ( A, U)) ∼ − → K i ( M ( S, U)) . (4.5) If furthermor e A is r e gular, then S is r e gular and ther e exist isomo rphisms K U i ( A ) ∼ − → K U i ( S ) . (4.6) Remark 4.4. If g = 0 a nd U is generated by the identit y , Theorem 4.3 reduces to a sligh tly w eaker vers ion of [27, Theorem 7]. See also the question raised in [27, p.118] (immediately b elo w [27, Theorem 6]). In order to pro v e t he theorem, we need some prelim inaries. Let z b e an indeter- minate, and consider the graded algebra S ′ = ⊕ p ≥ 0 ( F p S ) z p , where z is cen tral in S ′ and has degree 1. This is a subalgebra o f S [ z ]. W e endow S ′ with a U-action b y sp ecifying that z is U-inv arian t. This turns S ′ in to a U-mo dule algebra. Let S ′ + = ⊕ p> 0 ( F p S ) z p , t hen A = S ′ /S ′ + . Note also that S = S ′ /z S ′ . The ne xt result does not in v o lv e t he U-action. Lemma 4.5. Assume that S is left No etherian a n d A -ﬂat. (1) Then S ′ is left No etherian and A -ﬂat. (2) If it is further assume d that A (= S/S + ) has a ﬁnite pr oje ctive S -r esolution, then A (= S ′ /S ′ + ) also has a ﬁnite p r o je ctive S ′ -r esolution. Pr o of. F ilter S ′ b y letting F p S ′ consist o f polynomials in z with coeﬃcien ts in F p S . Then the asso ciated graded algebra of S ′ is giv en b y gr(S ′ ) = ⊕ p ≥ 0 F p S ′ F p − 1 S ′ = ⊕ p ≥ 0 S p z p . Since S is left No etherian, so also is S [ z ]. This then implies that S ′ is left No etherian (see, e.g., [2 7 , L emma 3.(i), p.119]). Giv en that S is A - ﬂat, so also is ev ery graded comp o nent S p , a nd in particular, F 0 S = S 0 . Corresp onding to an y short exact sequence 0 − → V ′ − → V − → V ′′ − → 0 of A -modules, w e ha v e the c omm utativ e diagram 0 0 0 ↓ ↓ ↓ 0 − → F p − 1 S ⊗ A V ′ − → F p − 1 S ⊗ A V − → F p − 1 S ⊗ A V ′′ − → 0 ↓ ↓ ↓ 0 − → F p S ⊗ A V ′ − → F p S ⊗ A V − → F p S ⊗ A V ′′ − → 0 ↓ ↓ ↓ 0 − → S p ⊗ A V ′ − → S p ⊗ A V − → S p ⊗ A V ′′ − → 0 ↓ ↓ ↓ 0 0 0 , where the columns are ob viously exact. Since the comp osition of the maps in the middle ro w is zero, exactness of the top and b ot t o m row s will imply the exactness of the middle o ne. Th us by induction on p , w e can sho w that F p S is A - ﬂat for all p . This then immediately leads to the A -ﬂatness of S ′ . 14 G. I. LEHRER AND R. B. ZHAN G T o pro ve the second part of the lemma, we note that S = S ′ /z S ′ leads to pd S ′ ( S ) = 1, where pd R ( M ) denotes the pro jectiv e dimension of the left R - mo dule M . Th us b y the Change Rings Theorem [32, Theorem 4.3.1], pd S ′ ( A ) ≤ pd S ( A ) + pd S ′ ( S ) < ∞ . This completes the pro of of t he lem ma.  Since S = S ′ / (1 − z ) S ′ , it follows from Lemma 4.5.(1) that S is left No etherian. Hence K i ( M ( S, U)) are deﬁned. The pro of of the followin g lemma requires b oth the Lo calisatio n Theorem [2 7, Theorem 5, p.105] and Deviss age Theorem [27, Theorem 4, p.104]. Lemma 4.6. Th e r e e x ists the fol lowing long exact se quenc e o f K-gr oups: · · · − → K 1 ( M g r ( S , U)) − → K 1 ( M g r ( S ′ , U)) − → K 1 ( M ( S, U)) − → K 0 ( M g r ( S , U)) − → K 0 ( M g r ( S ′ , U)) − → K 0 ( M ( S, U)) − → 0 . Pr o of. R ecall the crucial facts tha t z is U-in v a r ia n t and is also cen tral in S ′ . Let N b e the full sub category of M g r ( S ′ , U) consisting of mo dules killed b y some p o w er of z . This is a Serre sub category , so w e ma y deﬁne the quotient cate- gory M g r ( S ′ , U) / N . A more concrete wa y to view this construction is as fo l- lo ws. Let S ′ [ z − 1 ] b e the lo calisation of S ′ at z − 1 . Then the lo calisation func- tor M g r ( S ′ , U) − → M g r ( S ′ [ z − 1 ] , U) annihilates precisely the mo dules in N , and M g r ( S ′ , U) / N is e quiv alent to M g r ( S ′ [ z − 1 ] , U). F or a ny ob ject M in M g r ( S ′ [ z − 1 ] , U), z − 1 acts as a n isomorphism. Hence M is uniquely determined b y its degree 0 comp onen t. This leads to an equiv a lence of categories M g r ( S ′ [ z − 1 ] , U) ∼ = M ( S, U). Denote b y j : M g r ( S ′ , U) − → M ( S, U) the comp osition of t his equiv a lence with the lo calisation functor. Then j : M 7→ M / (1 − z ) M f or all M in M g r ( S ′ , U). No w using Quillen’s Lo calisation Theorem [27, Theorem 5, p.105], w e obtain (cf. op. ci t. p. 115 ) a long ex act seque nce of K-groups: · · · − → K i ( N ) − → K i ( M g r ( S ′ , U)) − → K i ( M ( S, U)) − → K i − 1 ( N ) − → · · · · · · − → K 0 ( M ( S, U)) − → 0 . Since S = S ′ /z S ′ , w e see that M g r ( S , U) is a full sub category of M g r ( S ′ , U) with ob jects the mo dules annihilated by z . Hence w e hav e an inclusion M g r ( S , U) ⊂ N . As an y ob ject N of N is annihilated by z k for some k , w e ha ve the ﬁltration 0 = z k N ⊂ z k − 1 N ⊂ · · · ⊂ z N ⊂ N with z i N z i − 1 N ob viously annihilated b y z f or eac h i . Hence the Devissage Theorem [27 , Theorem 4, p.112] applies to this situation, inducing isomor phisms of K-g r o ups K i ( N ) ∼ = K i ( M g r ( S , U)) fo r all i ≥ 0. Th e lemma follo ws.  Remark 4.7. Let i : M g r ( S , U) − → M g r ( S ′ , U) denote the comp osition o f the inclusions M g r ( S , U) ⊂ N ⊂ M g r ( S ′ , U). F rom the proo f of the le mma we see that maps b eside the connecting homomo r phism in the long exact sequence are i ∗ and j ∗ as indicated belo w · · · − → K i ( M g r ( S , U)) i ∗ − → K i ( M g r ( S ′ , U)) j ∗ − → K i ( M ( S, U)) − → . . . . EQUIV ARIANT K-THEOR Y OF QUANTUM GROUP A CTIONS 15 Pr o of of The or em 4.3. Since S satisﬁes the conditions of Theorem 4.2 , b y Lemma 4.5, S ′ also satisﬁes the conditions o f the theorem. Therefore, w e hav e the Z [ t ]- mo dule isomorphisms Z [ t ] ⊗ Z K i ( M ( A, U)) − → K i ( M g r ( S , U)) , 1 ⊗ g 7→ ( S ⊗ A − ) ∗ g , Z [ t ] ⊗ Z K i ( M ( A, U)) − → K i ( M g r ( S ′ , U)) , 1 ⊗ g 7→ ( S ′ ⊗ A − ) ∗ g . (4.7) Let us no w describ e the map i ∗ in Remark 4.7 more explicitly b y ﬁnding the map δ whic h renders the follo wing diagram comm utativ e: K i ( M g r ( S , U)) ✲ i ∗ K i ( M g r ( S ′ , U)) ✻ ∼ = ✻ ∼ = Z [ t ] ⊗ Z K i ( M ( A, U)) ✲ δ Z [ t ] ⊗ Z K i ( M ( A, U)). Here the v ertical isomorphisms are giv en by (4 .7). F or ev ery M in M ( A, U), we ha v e an exact s equence 0 − → S ′ ( − 1) ⊗ A M z − → S ′ ⊗ A M − → S ⊗ A M − → 0 in M g r ( S ′ , U), where S ⊗ A M is in M g r ( S , U) but is regarded as an ob ject of M g r ( S ′ , U) via the inclusion i . Therefore the comp osition i ◦ ( S ⊗ A − ) of functors M ( A, U) S ⊗ A − − → M ( S , U) i − → M g r ( S ′ , U) ﬁts into an exact sequence of functors 0 − → S ′ ( − 1) ⊗ A − − → S ′ ⊗ A − − → i ◦ ( S ⊗ A − ) − → 0 from M ( A, U) to M g r ( S ′ , U). By Corolla ry 1 o f Theorem 2 in [27, p.106] , we hav e i ∗ ◦ ( S ⊗ A − ) ∗ = ( S ′ ⊗ A − ) ∗ − ( S ′ ( − 1) ⊗ A − ) ∗ = (1 − t )( S ′ ⊗ A − ) ∗ . F rom this formula it is eviden t that the map δ is m ultiplicatio n b y 1 − t , whic h is injectiv e with cok ernel K i ( M ( A, U)). Therefore i ∗ is injectiv e with cokernel isomorphic to K i ( M ( A, U)). Using this information in the long exact sequence of Lemma 4 .6 , we deduce that the connecting morphism is zero, and K i ( M ( S, U)) is isomorphic to the cok ernel of i ∗ . Hence the comp osition of functors M ( A, U) − → M g r ( S ′ , U) j − → M ( S, U), M 7→ S ′ ⊗ A M 7→ S ⊗ A M (where S = S ′ / (1 − z ) S ′ ) ind uces an is omorphism K i ( M ( A, U)) − → K i ( M g r ( S ′ , U)) j ∗ − → K i ( M ( S, U)) . This pro v es the ﬁrs t assertion of the theorem. Giv en the conditio ns tha t S is left No etherian and A has ﬁnite pro jectiv e di- mension a s a left S -mo dule, [27, Lemma 4, p.120] a pplies and hence the regular- it y of A implies the regularit y of S . Th us it follow s fro m Prop osition 3.6 that K i ( M ( A, U)) = K U i ( A ) and K i ( M ( S, U)) = K U i ( S ) for a ll i . No w the second part of the theorem immediately follo ws from the ﬁrst part.  16 G. I. LEHRER AND R. B. ZHAN G 5. Quantum s ymmetric algebras W e no w apply results fro m Section 4 to compute the equiv ariant K-groups of a class of mo dule algebras o v er quantum groups. W e shall refer to these mo dule algebras as quantum s ymm etric algebr as ; these are quadratic algebras of Koszul t yp e naturally arising from the repre sen tat io n theory of quan tum groups. The quan tised co ordinate ring of aﬃne n -space is an example. See [19, 33, 8] for other examples. 5.1. Equiv arian t K -theory of quan tum symmetric algebras. Let V b e a ﬁnite dimensional v ector space ov er a ﬁeld k , and denote b y T ( V ) the tensor algebra o v er V . Give n a subset I of V ⊗ k V , w e denote b y h I i the tw o-sided ideal of T ( V ) generated b y I . D eﬁne the q uadr atic algebr a A := T ( V ) / h I i . W e shall also use the notatio n k { V , I } for A to indic ate the generating v ector space V and the deﬁning relations o f t he algebra explicitly . The alg ebra A is naturally Z + -graded sinc e h I i is. W e ha v e A = L ∞ i =0 A i , with A 0 = k a nd A 1 = V . W e shall say that a quadratic algebra A = k { V , I } is of PBW typ e if there ex ists a ba sis { v i | i = 1 , 2 , . . . , d } of V suc h that the elemen ts v a := v a 1 1 v a 2 2 · · · v a d d , with a := ( a 1 , a 2 , . . . , a d ) ∈ Z d + , form a basis (called the PBW basis) of A . Let k b e the ﬁe ld C ( q ) . Lemma 5.1. L et V b e a ﬁnite dimens ional mo dule o f typ e ( 1 , . . . , 1) over a quantum gr oup U . L et I ⊂ V ⊗ k V b e a U -submo dule. Then the quadr atic algebr a k { V , I } = T ( V ) / h I i is a U -mo dule algebr a. Pr o of. The tensor algebra T ( V ) has a natural U- mo dule algebra structure, with the U-action deﬁned by using the co-m ultiplication. Since I is a U- submo dule of V ⊗ V , so also is the t w o-sided ideal h I i . Hence A = T ( V ) / h I i is a U-mo dule algebra.  Deﬁnition 5.2. We c al l a U -mo dule alg e b r a A = k { V , I } of the typ e deﬁne d in L emma 5.1 a quantum symmetric algebra of the ﬁnite dimen sional U -mo dule V if it admits a PBW b asis. If A = k { V , I } admits a PBW basis, it is sometime s referred to as ‘ﬂat’. The next theorem is our main result concerning the equiv ariant K-theory o f quan- tum sy mmetric algebras. Theorem 5.3. L et A = k { V , I } b e a quantum symmetric algebr a of a ﬁnite di- mensional mo dule V ove r the quantum gr oup U . Assume that A is left No etherian, then K U i ( A ) = K i (U - mo d ) , for al l i = 0 , 1 , . . . , wher e U - mod is the c ate gory of ﬁnite dimensional left U -mo dules of typ e - (1 , . . . , 1) . W e shall pro v e this result using Theorem 4.3 of Sec tion 5.3. In order to do that, w e need some res ults from the theory of Ko szul algebras, which w e no w discuss . 5.2. Quadratic algebras and Koszul complexes. In this subsection, k ma y b e an y ﬁeld. Let V b e a ﬁnite dimensional vector space. Giv en a subspace I of V ⊗ k V , w e hav e the corresp onding quadratic algebra A = k { V , I } . Let V ∗ b e the dual vec tor space of V , and de ﬁne I ⊥ := { α ∈ V ∗ ⊗ k V ∗ | α ( w ) = 0 for all w ∈ I } . EQUIV ARIANT K-THEOR Y OF QUANTUM GROUP A CTIONS 17 This deﬁnition implicitly uses the canonical isomorphism ( V ⊗ k V ) ∗ ∼ = V ∗ ⊗ k V ∗ . Henceforth ⊗ will denote ⊗ k . Let h I ⊥ i b e the t w o-sided ideal of the tensor algebra T ( V ∗ ) o v er V ∗ generated b y I ⊥ . W e may deﬁne the quadratic a lgebra A ! := T ( V ∗ ) / h I ⊥ i , whic h is referred to as the dual quadr atic algeb r a of A . W e endow T ( V ∗ ) with a Z − := − Z + grading with V ∗ ha ving degree − 1. Then h I ⊥ i is a t wo-sided gra ded ideal, and A ! is Z − -graded with V ∗ ha ving degree − 1. Let A = k { V , I } and A ! = k { V ∗ , I ⊥ } b e dual quadratic algebras. Then A ! ⊗ A has a nat ura l algebra structure suc h that the subalgebras A and A ! comm ute. Let z b e the image of the iden tit y elemen t o f Hom k ( V , V ) in V ∗ ⊗ V under the natural isomorphism, and le t e A b e its imag e in A ! ⊗ A . It is easily veriﬁed tha t e 2 A = 0. W e regard A as a right A -mo dule, and A ! as a left A ! -mo dule. Then the graded dual A ! ∗ = L i ∈ Z + A ! ∗ i of A ! with A ! ∗ i = ( A ! − i ) ∗ has a nat ural righ t A ! -mo dule struc- ture. Hence A ! ∗ ⊗ A is a rig h t A ! ⊗ A -mo dule. The action of e A deﬁnes a diﬀerential on A ! ∗ ⊗ A , yielding the Koszul complex of A : · · · − → A ! 2 ∗ ⊗ A − → A ! 1 ∗ ⊗ A − → A. (5.1) Remark 5.4. W e may also regard A as a left A -mo dule and A ! as a right A ! -mo dule. Then w e ha ve a Koszul complex of left A -mo dules: · · · − → A ⊗ A ! 2 ∗ − → A ⊗ A ! 1 ∗ − → A, (5.2) where the diﬀeren tial is ˜ e A = P d i =1 v i ⊗ ¯ v i . F or an y pair of graded left A -mo dules M and N , o ne ma y compute the exten- sion spaces Ext • ( M , N ) := L i Ext i A ( M , N ) deﬁned as t he righ t deriv ed functor of the graded homomo r phism functor Hom A ( M , N ). Under the Y oneda pro duct, Ext • ( k , k ) f orms a gra ded a lgebra. A quadratic algebra A = k { V , I } is called Koszul if Ext • ( k , k ) ∼ = A ! as graded a lg ebras. If A is Koszul, so is also A ! . A ke y prop erty of a K o szul algebra is that the Koszul complex (5 .1) of A is a graded free resolution of the base ﬁeld k regarded as a right A -mo dule; that is, the complex · · · − → A ! 2 ∗ ⊗ A − → A ! 1 ∗ ⊗ A − → A ǫ − → k − → 0 (5.3) is exact. The map ǫ is the aug men tatio n A − → A/ A + , where A + = L i> 0 A i . F or the pro of of this fact, see e.g., [26, Corolla ry I I.3 .2 ]. Similarly , (5.2) leads to a graded free res olution for the bas e ﬁeld k regarded as a left A -mo dule in this case. F or quadrat ic algebras of PBW t yp e, w e ha ve the following result. Theorem 5.5. L et A = k { V , I } b e a quadr atic algebr a of PBW typ e, and denote by A ! its dual quadr a tic alg ebr a. Then: (1) The algebr a A i s Koszul. (2) L et d = dim k V , then dim k A i =  d + i − 1 i  and dim k A ! − i =  d i  for al l i ≥ 0 (3) The K oszul c o m plexes (5.1) and (5.2) of A ar e gr ade d fr e e r eso lutions of length dim k V of the b ase ﬁeld k . 18 G. I. LEHRER AND R. B. ZHAN G Pr o of. The K oszul nature of quadratic algebras of PBW ty p e is a w ell-known fact, whic h w as originally establishe d in [25, Theorem 5.3]. The dimension of A i can b e easily computed from the PBW basis. Let h A ( z ) = P ∞ i =0 z i dim k A i and h A ! ( z ) = P ∞ i =0 z i dim k A ! − i . Then h A ( z ) = 1 / (1 − z ) d . It follow s from some general facts on the Hilb ert series o f Ext • ( k , k ) that h A ( z ) h A ! ( − z ) = 1. By part (1), A is Ko szul, thus A ! ∼ = Ext • ( k , k ). Hence h A ! ( z ) = (1 + z ) d , whic h implies the claimed dimension form ula for A ! − i . The Koszul complexes a re free resolutions since A is Koszul by part (1). As A ! − i = 0 for all i > d , the length of the res olutions is dim k V .  Remark 5.6. Theorem 5.5.(3) will suﬃce for the purpose of pro ving Theorem 5.12 and Theorem 5.3 o n K -groups. The r esults b elo w giv e a direct pro of that (left) No etherian quan tum symmetric algebras are regular. Let M = L i ∈ Z M i b e a g raded mo dule for a quadratic algebra A = k { V , I } . If M is ﬁnitely generated, there exists some in t eger r suc h that M i = 0 for all i < r . That is, a ﬁnitely generated graded mo dule m ust be b ounded b elo w. Let M := A A + ⊗ A M ∼ = k ⊗ A M . The follo wing result, is a sp ecial case of [18, Theorem 4 .6]. Theorem 5.7. L et P b e a ﬁnitely gener ate d gr ade d pr oje ctive mo dule over a qua- dr atic algebr a A = k { V , I } . The n P is obtaine d fr om P by e x tension o f sc al a rs. That is, P ∼ = A ⊗ k P . Ther efor e, al l ﬁnitely gener ate d g r ad e d pr oje ctive m o dules over a quadr atic algebr a ar e fr e e. Recall that a complex of gra ded left A -mo dules · · · − → C n φ n − → C n − 1 φ n − 1 − → C n − 2 − → · · · is called minimal if φ n ( C n ) ⊆ A + C n − 1 for eac h n . A minimal r esolution is deﬁned similarly . Theorem 5.8. Every ﬁnitely gene r ate d gr ade d left mo dule ove r a quadr atic algebr a A = k { V , I } of PBW typ e has a minima l f r e e r e s o lution of length at most dim V . Pr o of. By [26 , Prop osition § 1.4.2.], eve ry ﬁnitely generated graded left A -mo dule M has a minimal free resolution F : · · · − → F 2 − → F 1 − → F 0 − → M − → 0 . (5.4) Th us it suﬃces to sho w that this resolution has ﬁnite length. Let us compute T or A • ( k , M ) from the complex k ⊗ A F . By using the minimality of F , we obtain T or A i ( k , M ) = k ⊗ A F i for all i. In particular, T or A i ( k , M ) = 0 if and only if F i = 0. On the other hand, w e may also compute T or A • ( k , M ) from the c omplex obta ined b y tensoring (ov er A ) the Koszul resolution of the base ﬁeld k (as a righ t A -mo dule) with M . By part (3) of Theorem 5.5, T or A i ( k , M ) = 0 for all i > dim V . This leads to F i = 0 for all i > dim V in the m inimal free resolution fo r M .  The follo wing result is a consequ ence of Theorem 5.8. EQUIV ARIANT K-THEOR Y OF QUANTUM GROUP A CTIONS 19 Theorem 5.9. Every ﬁ n itely gener a te d left m o dule o v er a quadr atic algebr a A = k { V , I } of PB W typ e has a fr e e r eso l ution of ﬁnite le ngth. Pr o of. L et A n φ − → A m − → M − → 0 b e an exact sequence of left A -mo dules. W e think o f A m and A n as consisting of row s with en tr ies fro m A . Then φ can b e represen ted as an n × m - matrix ( φ ij ) with en tries φ ij ∈ A , and a cts on A n b y matrix m ultiplication fr o m the righ t. No w consider the algebra T = A [ x ], whic h consists of p olynomials in x with co eﬃcien ts in A . W e stipulate that x comm utes with all elemen t s of A . Then A = T / (1 − x ) T . It is easy to see that T is a quadratic algebra whic h is also Koszul. Let r b e the smallest integer suc h t ha t ev ery entry φ ij of the ma t r ix of φ is con ta ined in A 0 ⊕ A 1 ⊕ · · · ⊕ A r . Then w e can write φ ij = φ ij [0] + φ ij [1] + · · · + φ ij [ r ] with φ ij [ k ] ∈ A k . Up on replacing the en tries φ ij of the matrix by φ ij ( x ) = x r φ ij [0] + x r − 1 φ ij [1] + · · · + φ ij [ r ], w e obta in a rectangular matr ix ˜ φ = ( ˜ φ ij ) with en tries whic h are homogeneous elemen t s of T o f de gree r . Regard ˜ φ as a left T - mo dule homomorphism T n − → T m , and let ˜ M = cok er ˜ φ . Then ˜ M is a graded T -mo dule, and A ⊗ T ˜ M = M . By Theorem 5.9 , w e ha v e a free res olution F o f ˜ M with ﬁnite le ngth. The homology of the complex A ⊗ T F is T or T • ( A, ˜ M ). T o compute T or T • ( A, ˜ M ), w e tensor with ˜ M the free resolution 0 − → T 1 − x − → T − → A − → 0 of the right T -mo dule A , obtaining 0 − → ˜ M 1 − x − → ˜ M . Since the a ction of 1 − x on any graded T -mo dule is inje ctiv e, the ab ov e comple x is exact. This sho ws that T or T i ( A, ˜ M ) = 0 , for all i ≥ 1 . (5.5) Note that A ⊗ T F is also a complex o f left A - mo dules, whic h are a ll free except for A ⊗ T ˜ M = M . F rom equation (5.5) w e see that this comple x is exact, th us is a free res olution of ﬁnite length for the A - mo dule M .  Recall t ha t a left mo dule E ov er a ring R is called stably fr e e if there exists a free left A -mo dule F of ﬁnite rank such that E ⊕ F ∼ = R n for some ﬁnite n . Clearly a stably free mo dule is ﬁnitely generated and pro jectiv e. It is easy to see that if a pro jectiv e mo dule admits a free r esolution of ﬁnite length, it is stably free. Con ve rsely , a pro jective R -mo dule E with a free r esolution 0 − → F n − → · · · − → F 1 − → F 0 − → E − → 0 of ﬁnite length n can b e sho wn to b e stably free b y a simple induction on n . Indeed, if n = 0, the claim is o bviously true. Let E ′ = K er ( F 0 − → E ), then w e ha ve the follo wing free resolution for E ′ : 0 − → F n − → · · · − → F 1 − → E ′ − → 0 . Since the length of the resolution is n − 1 , b y t he induction h yp othesis, E ′ is stably free. Hence E is stably free since F 0 ∼ = E ⊕ E ′ . The ne xt statemen t is an immediate consequence of Th eorem 5.9. 20 G. I. LEHRER AND R. B. ZHAN G Corollary 5.10. Every ﬁnitely gener ate d pr oje ctive m o dule over a quadr atic a lgebr a A = k { V , I } of PBW typ e is stably fr e e. Ev ery quadratic algebra A = k { V , I } of PBW type is a Z + -graded algebra with degree 0 subalgebra k . If the alg ebra is assumed t o b e left No etherian, then Theorem 5.9 implie s that it is regular. Lemma 5.11. A quad r atic algebr a of PBW typ e is left r e gular i f it is left No etherian. In p articular, eve ry left No etherian quantum symmetric alge b r a is left r e gular. W e ma y now use [27, Theorem 7] to compute the usual algebraic K-groups K i ( A ) of A . The result is as follo ws. Theorem 5.12. L et A b e a quadr atic algebr a of PBW typ e, and assume that A is left No etherian. Then K i ( A ) = K i ( k ) , i = 0 , 1 , . . . . In particular, K 0 ( k ) = Z . This is consis ten t with Corollary 5.10. 5.3. Pro of of Theorem 5.3. By Lemma 5.11 and Prop osition 3.6, K U i ( A ) = K i ( M ( A, U)). No w A = A 0 + A 1 + . . . is Z + -graded with A 0 = k . Th us w e ma y apply Theorem 4.2 to compute its U-equiv arian t K- groups. W e ha v e K U i ( A ) = K U i ( k ) = K i ( P ( k , U)) for all i = 0 , 1 , . . . . Note that M ( k , U) is t he category of ﬁnite dimensional left U-mo dules of type- (1 , 1 , . . . ). As is well-kno wn, M ( k , U) is semi-simple, thus P ( k , U) = M ( k , U) = U- mo d .  In particular K U 0 ( A ) is the Grothendiec k group of U- mod . 5.4. Examples. In this section, w e consider examples of quan tum symmetric al- gebras arising f rom natural mo dules for the quantum groups asso ciated with the classical series of Lie algebras. These quan tum symmetric alg ebras also feature prominen tly in the study of the in v a r ia n t theory of quan tum groups [19]. Example 5.13. Co or dinate algebr a of a quantum matrix. A familiar example o f quan tum symme tric a lgebras is O ( M ( m, n )), the co o rdinate a lgebra of a quan t um m × n matrix. It is generated b y x ij (1 ≤ i ≤ m , 1 ≤ j ≤ n ) sub j ect to t he fo llowing relations x ij x ik = q − 1 x ik x ij , j < k , x ij x k j = q − 1 x k j x ij , i < k , x ij x k l = x k l x ij , i < k , j > l , x ij x k l = x k l x ij − ( q − q − 1 ) x il x k j , i < k , j < l . (5.6) It is w ell kno wn that this is a mo dule algebra ov er U q ( sl n ) with a PBW basis con- sisting of ordered monomials of the elemen ts x ij . The U q ( sl n )-action on O ( M ( m, n )) can be described as follo ws. F or eac h i , the su bspace ⊕ n j =1 k x ij is isomorphic to the natural mo dule for U q ( sl n ). Thus O ( M ( m, n )) is a quadratic algebra of the U q ( sl n )- mo dule V whic h is the direct sum of m copies of the natural mo dule. In particular, when m = 1, all relations but the ﬁrst of (5.6) are v acuous, and w e obtain the q uan tised coo rdinate algebra of aﬃne n -space. EQUIV ARIANT K-THEOR Y OF QUANTUM GROUP A CTIONS 21 By [9, § I], O ( M ( m, n )) is left No etherian for all m and n . By The orem 5.12, the ordinary algebraic K-groups of O ( M ( m, n )) are giv en b y K i ( O ( M ( m, n ))) = K i ( k ) for all i . Theorem 5.3 also applies, and w e ha ve K U q ( sl n ) i ( O ( M ( m, n ))) ∼ = K i (U q ( sl n )- mo d ) , for all i. Example 5.14. Quantum symmetric algebr as asso ciate d with the natur al mo dules for U q ( so m ) and U q ( sp 2 n ) . W e ﬁrst brieﬂy recall the cons truction giv en in [19]. An imp o rtan t structural prop ert y of the quan tum g roup U = U q ( g ) a sso ciated with a simple Lie a lg ebra g is the braiding of its mo dule category provided by a univ ersal R -matrix [1 4]. W e may think of this as an inv ertible elemen t in some completion of U ⊗ k U, whic h satisﬁes the follo wing relations R ∆( x ) = ∆ ′ ( x ) R, ∀ x ∈ U , (∆ ⊗ id) R = R 13 R 23 , (id ⊗ ∆) R = R 13 R 12 , ( ǫ ⊗ id) R = (id ⊗ ǫ ) R = 1 ⊗ 1 , (5.7) where ǫ is the co-unit a nd ∆ ′ is the opp o site c o-multiplic ation. Here the subscripts of R 13 etc. ha ve the usual meaning as in [14]. It follow s from the second line of (5 .7) that R s atisﬁes the celebrated Y ang-Baxter equation R 12 R 13 R 23 = R 23 R 13 R 12 . Giv en a ﬁnite dimensional U-mo dule V , let R V , V denote t he a utomorphism of V ⊗ V deﬁned b y the univ ersal R -matrix of U. Let P : V ⊗ V − → V ⊗ V , v ⊗ w 7→ w ⊗ v , b e the p erm utation, and deﬁne ˇ R = P R V , V . Then ˇ R ∈ End U ( V ⊗ V ) by (5.7), and ˇ R has c haracteristic p olynomial of the form k + Y i =1  x − q χ (+) i  k − Y j =1  x + q χ ( − ) i  , where χ (+) i and χ ( − ) i are in tegers, and k ± and p ositiv e in tegers. Consider t he U- submo dule I − of V ⊗ V deﬁned b y I − = k + Y i =1  ˇ R − q χ (+) i  ( V ⊗ V ) . (5.8) W e ma y then form the U-mo dule algebra k { V , I − } asso ciated to V and I − . There is a classiﬁcation in [8, 35 ] of those irreducible U-mo dules V satisfying the condition that the corresp onding quadratic algebras k { V , I − } a dmit PBW bases. In part icular, the natural mo dules o f the quan tum groups asso ciat ed with t he classic al Lie algebras all ha v e this property [35]. Recall from [19] that if A and B are lo cally ﬁnite U-mo dule alg ebras, A ⊗ k B b ecomes a U-mo dule algebra if its m ultiplication is t wisted b y the univ ersal R - matrix. Explicitly , if w e write R = P t α t ⊗ β t , then for all a, a ′ ∈ A and b, b ′ ∈ B , ( a ⊗ b )( a ′ ⊗ b ′ ) = X t a ( β t · a ′ ) ⊗ ( α t · b ) b ′ . If C is a third lo cally ﬁnite U-mo dule algebra, t he U- mo dule algebras ( A ⊗ B ) ⊗ C and A ⊗ ( B ⊗ C ) are canonically isomorphic [19]. Therefore, giv en k { V , I − } asso ciated with an irreducible ﬁnite dimensional U- mo dule, we hav e lo cally ﬁnite U-mo dule algebras k { V , I − } ⊗ m for eac h p ositiv e integer m . F or a n y v ector space W we use the notation W n = ⊕ n W . 22 G. I. LEHRER AND R. B. ZHAN G Theorem 5.15. L et V b e the natur al mo dule of U q ( so m ) or U q ( sp 2 n ) , an d let I − b e the U -submo d ule of V ⊗ V d eﬁne d by ( 5 .8) . T hen S q ( V m ) := k { V , I − } ⊗ m is a No e therian quantum s ymm etric algebr a for ev e ry m . Here b y No etherian w e mean that the algebra is b oth left and right No etherian. T o prov e the t heorem, w e require some preliminaries. The following result f r o m [9] will b e of crucial imp o rtance. Lemma 5.16. [9, Prop o sition I.8.17] L et A b e an ass o ciative algebr a ov e r k . L et u 1 , u 2 , . . . , u N b e a ﬁnite se quenc e of elements which g ener ate A . Assume that ther e exist sc alars q ij ∈ k × , α st ij , β st ij ∈ k such that fo r al l i < j , u j u i = q ij u i u j + i − 1 X s =1 N X t =1  α st ij u s u t + β st ij u t u s  , (5.9) then A is No etherian. Observ e in particular that the algebra ˜ A presen ted in terms o f the generators u 1 , . . . , u N and the relations (5.9) is No etherian. Thus an y algebra deﬁned b y the same generators sub ject to ( 5 .9) and extra relations is a quotient of ˜ A , a nd hence is also No etherian. Let V b e the natural U q ( so 2 n )-mo dule, and let { v a | a = 1 , . . . , 2 n } b e a basis of w eight v ectors of V , with w eigh t s decreasing as a increases. Order this basis in the natural w ay : v 1 , v 2 , . . . , v 2 n . T hen k { V , I − } is generated b y v a (1 ≤ a ≤ 2 n ) with the follo wing relations v b v a = q v a v b , a < b ≤ 2 n, a + b 6 = 2 n + 1 , v n +1 v n = v n v n +1 , v 2 n − i v i +1 = q 2 v i +1 v 2 n − i − q v i v 2 n +1 − i + q v 2 n +1 − i v i , i ≤ n − 1 . (5.10) If we ignore the r elat io n arising from the i = n − 1 case of the third equation in (5.10), the remaining relations deﬁne an algebra whic h satisﬁes the conditions of Lemma 5.1 6. Therefore, k { V , I } is No etherian. It is know n [8, 35] that k { V , I } admits a PBW basis, and hence is a No etherian quan tum symmetric algebra. W e now turn to The orem 5.15. Pr o of of The or em 5.15. It w as sho wn in [19] t hat S q ( V m ) a dmits a PBW basis for ev ery m , whence it suﬃces to prov e that this algebra is No etherian. If V is the natural U q ( so 2 n )-mo dule, w e may regard S q ( V m ) as g enerated by X ia with 1 ≤ i ≤ m and 1 ≤ a ≤ 2 n , sub ject to t he relations R(1) and R(2) b elo w: R(1): F or an y i , X ib X ia = q X ia X ib , a < b ≤ 2 n, a + b 6 = 2 n + 1 , X i,n +1 X in = X in X i,n +1 , X i, 2 n − s X i,s +1 = q 2 X i,s +1 X i, 2 n − s − q X is X i, 2 n +1 − s + q X i, 2 n +1 − s X is , s ≤ n − 1 . R(2): F or i < j a nd a ll a, b , X j b X ia = q − 1 ab X ia X j b + X t ( β ′ t · X ia )( α ′ t · X j b ) , where q ab is q − 1 if a = b , is q if a + b = 2 n + 1, a nd is 1 otherwise. EQUIV ARIANT K-THEOR Y OF QUANTUM GROUP A CTIONS 23 Here P a k X ia ∼ = V as U-mo dule for eac h i , and w e use the form R = K + P t α ′ t ⊗ β ′ t for the univ ersal R -matrix, where K a cts by K ( X j b ⊗ X ia ) = q − 1 ab X j b ⊗ X ia . Actions of α ′ t (resp. β ′ t ) increase (resp. decrease) w eights . W e ha ve β ′ t · X ia = ζ at X ia t and α ′ t · X j b = η tb X j b t for some a t > a and b t < b , where ζ ta and η tb are scalars suc h that ζ ta η tb 6 = 0 only for ﬁn itely man y t . Order the elemen ts X ia as fo llo ws: X m 1 , X m 2 , . . . , X m, 2 n ; X m − 1 , 1 , X m − 1 , 2 , . . . , X m − 1 , 2 n ; . . . ; X 11 , X 12 , . . . , X 1 , 2 n . Note that relat io ns R(1) are the same as (5.10) with v a replaced b y X ia , and the order of the elemen ts agrees with that of the v a . The relations R(2 ) ma y b e re- written as X ia X j b = q ab X j b X ia − q ab X t ( β ′ t · X ia )( α ′ t · X j b ) = q ab X j b X ia − q ab X t ζ at η tb X ia t X j b t for i < j a nd all a, b . T hese relations are in t he form of (5.9 ) . Therefore S q ( V m ) meets the conditions of Lemma 5.16, and hence is No etherian. This completes t he pro of for t he cas e of U q ( so 2 n ) When V is the natural U q ( so 2 n +1 )-mo dule or natural U q ( sp 2 n )-mo dule, there are deﬁning relations of S q ( V m ) analogous t o those in the U q ( so 2 n ) case [19], and similar reasoning shows that S q ( V m ) is also No etherian. W e leav e the details of the pro o f in these cases to the reader.  In view of Theorem 5.15, w e may now apply Theorem 5.12 and Theorem 5.3 to compute the o rdinary and equiv ar ian t K -groups of S q ( V m ). W e ha v e K i ( S q ( V m )) ∼ = K i ( k ) , K U q ( g ) i ( S q ( V m )) ∼ = K i (U q ( g )- mo d ) , f or all i, where U q ( g ) is U q ( so m ) or U q ( sp 2 n ). Example 5.17. W e conside r the W eyl algebra W q of degree 1 ov er k = C ( q ). It is generated b y x, y sub ject to the re lation xy − q − 1 y x = 1 . This is a mo dule algebra ov er U q ( sl 2 ) if w e iden tify { x, y } with the standard basis of the natural U q ( sl 2 )-mo dule k 2 . Let F i W q b e the span of the elemen ts x j − t y t with i ≥ j ≥ t ≥ 0. Then w e hav e a complete ascending ﬁltra tion 0 ⊂ F 0 W q ⊂ F 1 W q ⊂ F 2 W q ⊂ . . . , whic h is stable under the U q ( sl 2 )-action. The asso ciated graded algebra g r ( W q ) is the U q ( sl 2 )- mo dule algebra generated b y x, y sub ject to the relation xy = q − 1 y x . By Lemma 5.11, this algebra is left re gular. Therefore Theorem 5.1 2 and Theorem 5.3 apply to W q , w e obtain K i ( W q ) = K i ( k ) and K U q ( sl 2 ) i ( W q ) = K i ( P ( k , U q ( sl 2 ))) ( i ≥ 0) fo r the Quillen K-groups a nd equi- v ar ia n t K -groups resp ectiv ely . In the presen t case, M ( k , U q ( sl 2 )) = P ( k , U q ( sl 2 )) = U q ( sl 2 )- mo d is the category of ﬁnite dimensional U q ( sl 2 )-mo dules o f t yp e- (1 , . . . , 1). Hence K U q ( sl 2 ) i ( W q ) = K i (U q ( sl 2 )- mo d ) for all i ≥ 0. 24 G. I. LEHRER AND R. B. ZHAN G 6. Quantum homogeneous sp aces Quan tum homogeneous spaces [15] are a class of noncomm utativ e geometries with quan tum group symmetries, whic h ha v e b een widely studied (see, e.g., [11, 20] and the referenc es therein). W e shall dev elop their equiv arian t K-theory in this s ection. W e men tio n that the equiv ariant K 0 -groups of quantum homogeneous spaces ha v e b een determined in [34]. 6.1. Quan tum homogenous spaces. W e recall fro m [15, 34] some bac kground material o n quantum homogeneous spaces, whic h will b e needed lat er. Let V b e an ob ject in U- mo d , and let π : U − → End k ( V ) b e the corresp onding matrix represen tation of U relativ e t o some basis of V . Then t here exist elemen ts t ij ( i, j = 1 , 2 , . . . , dim V ) in the dual U ∗ of U suc h that for any x ∈ U, we ha v e π ( x ) ij = h t ij , x i for all i, j . The t ij will b e referred to as the co ordinate functions of the ﬁnite dimensional represen ta tion π . It follo ws from standard facts in Hopf algebra theory [23] that the co ordinate functions of all the v a r io us U-represen tations asso ciated with the U-mo dules in U- mo d span a Hopf algebra O q (U), whic h is a Hopf subalgebra of the ﬁnite dual of U (see [23] fo r this notion). W e s hall denote t he co-m ultiplication and the a ntipo de of O q (U) by ∆ and S resp ectiv ely , but the co- unit will b e denoted b y ǫ 0 . Note that the co-unit (resp. unit) of U b ecomes the unit (resp. co-unit) o f O q ( U ). W e shall denote O q (U q ( g )) by A g for brevit y . There exist t w o natural a ctio ns R and L of U o n A g [15], whic h correspond to left and right translation in the con text of Lie groups. These actions are resp ectiv ely deﬁned b y R x f = X ( f ) f (1) < f (2) , x >, L x f = X ( f ) < f (1) , S ( x ) > f (2) for all x ∈ U and f ∈ A g , where as f ( x ) = h f , x i for an y x ∈ U and f in the ﬁnite dual of U. These tw o actions clearly comm ute by the coasso ciativit y o f t he Hopf structure on A g . It can b e sho wn that A g forms a U-mo dule algebra under both actions. How eve r, some care needs to b e exercised in the case o f L , a s fo r an y f , g ∈ A g and x ∈ U, w e ha v e L x ( f g ) = X ( f ) , ( g ) h f (1) g (1) , S ( x ) i f (2) g (2) = X ( f ) , ( g ) , ( x ) h f (1) , S ( x (2) ) ih g (1) , S ( x (1) ) i f (2) g (2) = X ( x ) L x (2) ( f ) L x (1) ( g ) . This shows that under the action L , A g forms a mo dule alg ebra ov er U ′ , whic h is U, with the opp osite co-m ultiplication ∆ ′ . Let Θ b e a subset of { 1 , 2 , . . . , r } , where r is the rank o f g . W e denote by U q ( l ) the Hopf subalgebra of U generated b y the elemen ts of { k ± i | 1 ≤ i ≤ r } ∪ { e j , f j | j ∈ Θ } . W e denote b y U q ( l )- mo d the catego ry of ﬁnite dimensional left U q ( l )-mo dules of t yp e-(1 , . . . , 1). This category is semisimple. EQUIV ARIANT K-THEOR Y OF QUANTUM GROUP A CTIONS 25 F ollo wing [15], w e deﬁne A = { f ∈ A g | L x ( f ) = ǫ ( x ) f , ∀ x ∈ U q ( l ) } = A U q ( l ) g . (6.1) This is the submo dule of U q ( l )-in v arian ts of A g . The follo wing result is fairly straigh tforw ard (cf. [15, 34]). Theorem 6.1. The subsp ac e A forms a lo c al ly ﬁnite U -mo dule algebr a under the action R . F urthermor e, A is (b oth left and right) No etherian. Indeed, since U q ( l ) is a Hopf subalgebra of U, it follow s fr o m the deﬁnition that A is a subalgebra of A g . Since left a nd right t r a nslations comm ute, the U- mo dule algebra structure o f A g under R descends to A . Being a subalgebra of A g whic h is con tained in the ﬁnite dual of U, A m ust b e lo cally ﬁnite under the U-action R . The fact that A is No etherian is pro v ed in [34]. It was sho wn in [15] tha t t he algebra A is the natural quantum ana logue o f the algebra of complex v alued (smoo t h) functions on the real manifold G/K for a com- pact connected Lie group G a nd a closed subgroup K , where the Lie algebras L ie ( G ) and L ie ( K ) ha v e complexiﬁcations g and l resp ectiv ely . Th us the noncomm uta tiv e space determined by the algebra A is referred t o as a quantum homo g ene ous sp ac e in [15]. Remark 6.2. A quan tum homogeneous space de ﬁned this w ay is a quan tisatio n of the real manifold underlying a compact homogeneous space. A complex structure (in a generalised sense) o n the quantum homogeneous space was discussed and used in establis hing Borel-W eil t yp e theorems in [15]. 6.2. Equiv arian t K -theory of quan tum homogen eous spaces. F or an y ob ject Ξ of U q ( l )- mo d , w e deﬁne the induced U-mo dule as follo ws: (6.2) S ( Ξ) :=    ζ ∈ Ξ ⊗ A g       X ( x ) ( x (1) ⊗ L x (2) ) ζ = ǫ ( x ) ζ , ∀ x ∈ U q ( l )    = (Ξ ⊗ k A g ) U q ( l ) , where U q ( l ) acts on the tens or pro duct via (id ⊗ L ) ◦ ∆. Then S (Ξ) is b ot h a left A -mo dule and left U-mo dule with A and U actions deﬁned, for b ∈ A , x ∈ U and ζ = P v i ⊗ a i ∈ S (Ξ), by bζ = X v i ⊗ ba i , (6.3) xζ = (id Ξ ⊗ R x ) ζ = X v i ⊗ R x ( a i ) . (6.4) W e hav e x ( bζ ) = P v i ⊗ R x ( ba i ) = X ( x ) R x (1) ( b )( x (2) ζ ) , for b ∈ A. Th us S (Ξ) indeed forms a U-equiv aria n t A -mo dule. The follo wing results w ere prov ed in [15, 34]. Theorem 6.3. (1) L et V b e the r e s triction of a ﬁnite dimensional U -mo dule to a U q ( l ) -mo dule. Then S ( V ) ∼ = V ⊗ k A in M ( A, U) . (2) F or any obje ct Ξ in U q ( l ) - mo d , S (Ξ) is an obje c t of P ( A, U) . 26 G. I. LEHRER AND R. B. ZHAN G Recall fro m [15] that par t (2) of t he theorem follow s fro m part (1). Indeed, Ξ can alwa ys b e embedded in the restriction of some ﬁnite dimensional U-mo dule as a direct summand. That is, there exist a ﬁnite dimensional U-mo dule W and a U q ( l )-mo dule Ξ ⊥ suc h that W ∼ = Ξ ⊕ Ξ ⊥ as U q ( l )-mo dule. It follow s f r om part (1) of Theorem 6.3 that S (Ξ) ⊕ S (Ξ ⊥ ) ∼ = W ⊗ A . By Corollary 3.4, S (Ξ) is in P ( A, U). W e extend (6.2) to a co v ariant functor S : U q ( l )- mo d − → P ( A, U) , (6.5) whic h acts on ob jects of U q ( l )- mo d a ccording to (6.2) and sends a morphism f to f ⊗ id A g . Since U q ( l )- mo d is semi-simple and S ( V ⊕ W ) = S ( V ) ⊕ S ( W ) for an y direct sum V ⊕ W o f ob jects in U q ( l )- mo d , the functor S is ex act. Let I = { f ∈ A | f ( 1 ) = 0 } ; this is a maximal tw o -sided ideal of A . W e hav e A/I ∼ = k . F or an y x ∈ U q ( l ) and a ∈ I , h R x ( a ) , 1 i = ǫ ( x ) a (1) = 0, thus I forms a U q ( l )-algebra under the r estriction of the action R . This implies that for an y U-equiv ariant A -mo dule M , I M is a U q ( l )-equviarain t A -submo dule of M . This can b e seen from the follow ing calculation: for any a ∈ I and m ∈ M , w e hav e x ( am ) = P ( x ) R x (1) ( a ) x (2) m ∈ I M for all x ∈ U q ( l ). Giv en a U- equiv ariant A -mo dule M , let M 0 = M /I M ; this is a U q ( l )-equiv a r ia n t A -mo dule in w hic h a ∈ A acts as a (1) ∈ k . Denote the natura l surjection by p : M − → M 0 . (6.6) This is an A -U q ( l )-linear map. F or a n y ob ject M in M ( A, U), w e can ﬁnd a ﬁnite dimensional U-submo dule W whic h generates M . By Lemma 3.3, the A -U-map A ⊗ k W − → M , a ⊗ w 7→ aw , is surjectiv e. Therefore M 0 = p ( W ). Note that W is semi-simple as U-mo dule and hence also as U q ( l )-mo dule. Th us M 0 b elongs to U q ( l )- mo d . W e therefore ha v e a co v ariant functor E : M ( A, U) − → U q ( l )- mo d , whic h sends an ob ject M in M ( A, U) to M 0 , and is deﬁned on morphisms in the ob vious w a y . W e ma y restrict this f unctor to the full sub category P ( A, U) to obtain a co v ariant functor E P : P ( A, U) − → U q ( l )- mo d . (6.7) Theorem 6.4. The functors S : U q ( l ) - mo d − → P ( A, U) and E P : P ( A, U) − → U q ( l ) - mo d r esp e ctively deﬁne d by (6.5) and (6.7) ar e mutual ly inverse e quivalen c es of c ate gories. W e will pro v e the theorem in Section 6.3. It has the follo wing conse quence. Corollary 6.5. Ther e is an isomorph ism of ab e lian gr oups K U i ( A ) ∼ = K i (U q ( l ) - mo d ) for e ach i ≥ 0 , whe r e U q ( l ) - mo d is the c ate gory of ﬁ nite dim e nsional left U q ( l ) - mo dules of typ e- (1 , . . . , 1) (which is semi-si mple). This implies in particular that K U 0 ( A ) is isomorphic to the Grot hendiec k group of U q ( l )- mo d , a re sult pro v ed in [15]. EQUIV ARIANT K-THEOR Y OF QUANTUM GROUP A CTIONS 27 6.3. Pro of of Theorem 6.4. W e no w prov e Theorem 6.4 b y means of a series of lemmas. F or any V in U q ( l )- mo d , S ( V ) is the subspace of U q ( l )-in v arian ts in V ⊗ k A g with resp ect to the action id V ⊗ L U q ( l ) . The linear map V ⊗ k A g − → V giv en by ζ = X i v i ⊗ f i 7→ ζ (1) = X i f i (1) v i , induces a linear map (‘ev aluation’) ev : S ( V ) − → V , ζ 7→ ζ (1 ) . (6.8) Note that S ( V ) is an A -U q ( l )-mo dule with the standar d actions deﬁned by (6.3) and the restriction of (6.4). W e ma y a lso deﬁne an A -mo dule structure on V in whic h each a ∈ A acts as scalar m ultiplication b y a (1). This mak es V in to an A -U q ( l )-mo dule. Lemma 6.6. A ny V in U q ( l ) - mo d may b e r e gar de d as an A - U q ( l ) -mo dule as ab ove. The map (6.8) is A - U q ( l ) -line ar. Pr o of. G iv en its imp ortance, w e g ive a brief pro of of this lemma. F or an y ζ ∈ S ( V ) and u ∈ U q ( l ), w e ha ve X ( u ) ( u (1) ⊗ L u (2) ) ζ ⊗ u (3) = ζ ⊗ u b y the U q ( l )-in v ariance of S ( V ). Using P ( u ) ( u (1) ⊗ L u (2) ) ζ ( u (3) ) = u · ev ( ζ ) and ζ ( u ) = ev ((id V ⊗ R u ) ζ ), we obtain u · ev ( ζ ) = ev ((id V ⊗ R u ) ζ ). F ina lly , for an y a ∈ A , w e ha ve ev ( aζ ) = a (1) ev ( ζ ). This completes the pro of.  In view of the lemm a and the fact tha t the map (6 .6) is U q ( l )-linear, we hav e the follo wing U q ( l )-map for each V : ǫ V : E ◦ S ( V ) − → V , p ( ζ ) 7→ ev ( ζ ) = ζ ( 1 ) . (6.9) Prop osition 6.7. T he map ǫ V deﬁne d by (6.9) for e ach obje ct V in U q ( l ) - mo d gives rise to a natur al tr an sformation ǫ : E ◦ S − → id U q ( l ) - mo d , (6.10) which is in fact a n atur al isomorphis m . Pr o of. F or an y map α : V − → V ′ in U q ( l )- mo d , E ◦ S ( α ) is giv en b y E ◦ S ( α ) ( p ( ζ )) = p (( α ⊗ id A ) ζ ) , ∀ p ( ζ ) ∈ E ◦ S ( V ) . No w ǫ V ◦ p (( α ⊗ id A ) ζ ) = (( α ⊗ id A ) ζ )(1) = α ( ζ (1)). This prov es the comm utativit y of the following diagr am E ◦ S ( V ) ǫ V − → V ↓ ↓ α E ◦ S ( V ′ ) ǫ V ′ − → V ′ , where the left v ertical ma p is E ◦ S ( α ). Hence ǫ is a nat ur a l transformatio n b et w een the functors E ◦ S a nd id U q ( l )- mo d . W e ha ve already noted that any mo dule V in U q ( l )- mo d may b e em b edded in the restriction of a ﬁnite dimensional U-mo dule W as a direct summand, t hat is, 28 G. I. LEHRER AND R. B. ZHAN G W ∼ = V ⊕ V ⊥ for some U q ( l )-mo dule V ⊥ . No w S ( W ) ∼ = W ⊗ k A as A − U-mo dule, b y Theorem 6 .3(2). Using this isomorphism, w e obtain that E ◦ S ( W ) ∼ = W , ev ( S ( W )) = W , and that ǫ W : E ◦ S ( W ) − → W is an isomorphism. Since S ( V ) ⊕ S ( V ⊥ ) ∼ = W ⊗ k A , w e hav e ǫ W = ǫ V ⊕ ǫ V ⊥ . As ǫ W is a n isomorphism, b oth ǫ V and ǫ V ⊥ a r e isomorphisms.  It follows from App endix A that there is a one to one corresp o ndence b et w een lo cally ﬁnite left U-mo dules (of type-(1 , 1 , . . . , 1)) a nd right A g -como dules. F or an y lo cally ﬁnite left U-mo dule M , denote the correspo nding right A g -como dule structure map b y δ M : M − → M ⊗ k A g , w 7→ δ M ( w ) = X ( w ) w (1) ⊗ w (2) . Let M b e and ob ject of M ( A, U), and consider the comp osition M δ M − → M ⊗ k A g p ⊗ id − → p ( M ) ⊗ k A g . No t e that for an y m ∈ M and u ∈ U q ( l ), w e hav e X p ( u (1) m (1) ) ⊗ L u (2) ( m (2) ) = X p ( m (1) ) ⊗ h m (2) , u (1) ih m (3) , S ( u (2) ) i m (4) = ǫ ( u ) X p ( m (1) ) ⊗ m (2) . Hence the image of this map is con tained in S ( E ( M )), and w e obtain a map ˆ δ M = ( p ⊗ id) ◦ δ M : M − → S ( E ( M )) , whic h is clearly A -linear. Moreov er for an y m ∈ M a nd x ∈ U, w e ha ve ˆ δ M ( xm ) = X p ( m (1) ) ⊗ R x m (2) = x ˆ δ M ( m ) . Th us the map is also U-linear, and is t herefore a ho mo mo r phism of A − U-mo dules. This leads to the fo llo wing result, whic h w as pro v ed in [3 4, Prop osition 3 .8 ] in a sligh tly diﬀeren t form, and w as used there to sho w that the Gro thediec k groups of P ( A, U) and U q ( l )- mo d are isomorphic. Lemma 6.8. If M is in P ( A, U) , then ˆ δ M : M − → S ◦ E ( M ) is an isomorphism of A − U -mo dules. Prop osition 6.9. The maps ˆ δ M , for M in M ( A, U) deﬁne a natur al tr ansf o rmation id M ( A, U) − → S ◦ E . Pr o of. L et β : M − → N b e a m orphism in M ( A, U). Then S ◦ E ( β ) ˆ δ M = ( p N ⊗ id)( β ⊗ id) δ M , where p N is the m ap (6.6) for N . Using t he fact that ( β ⊗ id) δ M = δ N β , w e obtain S ◦ E ( β ) ˆ δ M = ˆ δ N β . That is, the follow ing diagram c omm utes M ˆ δ M − → S ◦ E ( M ) β ↓ ↓ N ˆ δ N − → S ◦ E ( N ) , where the right vertical map is S ◦ E ( β ).  The ne xt statemen t is an immediate consequence of Lemm a 6.9 and Lemma 6.8. EQUIV ARIANT K-THEOR Y OF QUANTUM GROUP A CTIONS 29 Prop osition 6.10. Ther e is a n atur al isomorphis m ˆ δ : id P ( A, U) − → S ◦ E P . Pr o of of The or em 6.4. It follow s from Prop osition 6.7 and Prop o sition 6 .10 that the categories U q ( l )- mo d and P ( A, U) are equiv alent.  Appendix A. Comodules and smash products The notions of mo dule algebras and equiv a r ian t mo dules can also b e formulated in terms o f co-algebras and como dules, whic h w e discuss brieﬂy here. Some of this material is used in Section 6 .3. W e shall a lso discuss the relationship b et we en lo cally ﬁnite equiv ariant mo dules ov er a U-mo dule algebra A , a nd ﬁnitely generated mo dules o ver the smash pro duct of A and U. Recall from Section 6.3 the co ordinate functions of the U-represen tations a sso ci- ated with the U-mo dules in U- mo d span a Hopf algebra O q (U). A como dule o v er O q ( U ) is a v ector space M with a k -linear map δ : M − → M ⊗ k O q ( U ) satisfying the following conditions: ( δ ⊗ id O q ( U ) ) ◦ δ = (id M ⊗ ∆) δ, (id M ⊗ ǫ 0 ) δ = id M . W e use Sw eedler’s notation δ ( v ) = P ( v ) v (1) ⊗ v (2) for the co-action on v ∈ M , so that v (1) ∈ M , while v (2) ∈ O q ( U ). A lo cally ﬁnite left U-mo dule M of type-(1 , . . . , 1 ) is natura lly a right como dule o v er O q ( U ), a nd vice v ersa. The como dule structure map δ : M − → M ⊗ k O q ( U ) and the mo dule s tructure map φ : U ⊗ M − → M a re related to eac h other b y δ ( v )( x ) = φ ( x ⊗ v ) , for all x ∈ U , v ∈ M . In this deﬁnition, lo cal U-ﬁniteness of M is needed in order for δ ( v ) to lie in M ⊗ O q (U) for all v ∈ M . If a U- mo dule is not lo cally ﬁnite, it do es not corresp ond to any como dule ov er O q (U). The lo cal ﬁniteness condition is built in to the deﬁnition of como dules b ecause O q ( U ) is the Hopf algebra spanned b y the co ordinate functions of U-r epresen tations corresp onding to ob jects in U- mo d . No w a lo cally ﬁnite U-mo dule algebra A of type-(1 , . . . , 1 ) is nothing but an asso ciativ e alg ebra whic h is a righ t como dule o v er O q (U) satisfying the conditions δ A (1 A ) = 1 A ⊗ ǫ, δ A ( ab ) = X ( a ) , ( b ) a (1) b (1) ⊗ a (2) b (2) , ∀ a, b ∈ A, where δ A is the como dule structure map of A . Let A b e a lo cally ﬁnite U-mo dule algebra of t yp e- ( 1 , . . . , 1), and as ab o v e, denote the co-action of O q (U) on any a ∈ A b y a 7→ P ( a ) a (1) ⊗ a (2) . A lo cally U-ﬁnite, type-(1 , . . . , 1) , U-equiv a rian t A -mo dule M is a left A -mo dule, whic h is also a righ t O q (U)-como dule, such that the A -mo dule structure and O q (U)- como dule s tructure δ M : M − → M ⊗ O q (U) are c ompatible in the sense that for all a ∈ A and v ∈ M , δ M ( av ) = X ( a ) , ( v ) a (1) v (1) ⊗ a (2) v (2) . A notion closely related to equiv arian t mo dules is the smash pr o d uct [23, Deﬁnition 4.1.3]. Giv en a U-mo dule algebra A , one ma y construct the smash pro duct R := 30 G. I. LEHRER AND R. B. ZHAN G A #U, which is an asso ciative algebra with underlying v ector space A ⊗ k U and m ultiplication deﬁned b y ( a ⊗ u )( b ⊗ v ) = X ( u ) a ( u (1) · b ) ⊗ u (2) v for all a, b ∈ A and u, v ∈ U. It is easy to show that a left U-equiv arian t A -mo dule is in fa ct a left R -mo dule. Ho wev er, a ﬁnitely generated R -mo dule need not b e lo cally U-ﬁnite, a nd so is not generally in M ( A, U). Therefore, P ( A, U) is diﬀerent from the category of ﬁnitely generated pro jectiv e R - mo dules. A case in p oin t is the following example. T ake A = C ( q ) b e equipp ed with trivial U-action (thro ugh the co-unit) . In this case, b oth M ( A, U) and P ( A, U) coincide with U- mo d . On the o ther hand, the smash pro duct R = A #U is U itself. The category of ﬁnitely generated pro jectiv e U-mo dules is totally diﬀeren t from the category U- mod . This sho ws that the equiv ariant K-theory of the U-mo dule algebra A intro duced in Section 2.2 is quite diﬀerent from the usual K-theory of the smash pro duct R := A #U, a fa ct whic h w e hav e already pointed out in Rem ark 2.6. Ac kno wledgemen ts . W e thank Ngau Lam for discussions throughout the course of this w ork. W e also thank Alan Carey for helpful corresp ondence and Pete r D ono v a n for suggestions on impro ving the ﬁrst draft of this pap er. This w ork is supp orted b y the Australian Researc h Council. 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