Derived Categories and Morita Duality Theory

Jun-ichi Miyachi Department of Mathematics, Tokyo Gakugei University, Koganei-shi, Tokyo, 184, Japan

Abstract

We define a cotilting bimodule complex as the non-commutative version of a dualizing complex, and show that a cotilting bimodule complex includes all indecomposable injective modules in case of Noetherian rings. Moreover we define strong-Morita derived duality, and show that existence of a cotilting bimodule complex is equivalent to one of strong-Morita derived duality.

Introduction

In algebraic geometry, the notion of dualizing complexes was introduced by Grothendieck and Hartshorne [4], and was studied by several authors. They had started to use technique of local duality, and used developed technique of duality for derived categories [4]. Yekutieli developed this theory to deal with case of non-commutative graded k- algebras [13]. In representation theory, Rickard gave a 'Morita theory' for derived categories of categories [10]. He also introduced tilting bimodule complexes in case of projective k- algebras over a k , and studied the relations between tilting bimodule complexes and derived equivalences [11]. Afterward several authors in representation theory studied derived categories of module categories (for example [5] and [7]). We studied cotilting bimodules as the non-commutative ring version of dualizing modules, and the conditions that bimodules induce a localization duality of derived categories [8]. The purpose of this paper is to study a 'Morita duality theory' for derived categories in case of coherent rings, that is, the relations between cotilting bimodule complexes and dualities for derived

1 categories. From the point of view of dualizing complexes, this notion is also the non- commutative ring version of dualizing complexes. In section 2, we study bimodule complexes which induce localization dualities of derived categories of modules (Theorem 2.6 and Corollary 2.7), and show that a cotilting bimodule complex induces a Morita derived duality (Corollary 2.8). Moreover, we show a cotilting bimodule complex is a finitely embedding cogenerator, and in case of Noetherian rings, a cotilting bimodule complex includes every injective indecomposable module (Theorem 2.9, Corollary 2.10, 2.10 and 2.11). This property is also the non-commutative ring version of residual complexes in algebraic geometry. For an algebra A over a commutative Noetherian ring R, we construct a dualizing A-bimodule complex by using an R -module dualizing complex (Theorem 2.14 and Corollary 2.15). In section 3, in case of projective k- algebras over a commutative ring k , we give a 'Morita duality theorem' for derived categories (Theorem 3.3 and Corollary 3.6). As well as the uniqueness of the dualizing complex, for local rings, we have the uniqueness of the cotilting bimodule complex (Proposition 3.7). Throughout this paper, we assume that all rings have non-zero unity, and that all modules are unital.

1. Preliminaries

Let G : U ® V and F : V ® U be contravariant ¶-functors between triangulated categories. We call G continuous if G sends direct sums to direct products (if they exist).

We call {G, F} a right adjoint pair if there is a functorial isomorphism HomU(X, FY ) @

HomV(Y, GX ) for all X Î U and Y Î V. It is easy to see that if {G, F} is a right adjoint pair, then G and F are continuous. We call {V; G, F} a localization duality of U provided that {G, o F} is a right adjoint pair, and that the natural morphism idV ® G F is an isomorphism (see [8]).

Let A be an additive category, K (A) a homotopy category of A, and K +(A), K -(A) and K b(A) full subcategories of K (A) generated by the bounded below complexes, the bounded above complexes, the bounded complexes, respectively. For a full subcategory B of an A, let K *,b(B) be the full subcategory of K *(B) generated by

2 complexes which have bounded homologies, and K *(B)Qis the quotient category of K *(B) by the multiplicative set of quasi-isomorphisms, where * = + or -. We denote K*(A)Qis by

* D*(A). For a thick abelian subcategory C of A, we denote by DC (A) a full subcategory of D *(A) generated by complexes of which all homologies belong to C (see [4] for details).

¥ i For a complex X := (X , d i), we define the following truncations:

¥ n+1 n+2 s>n(X ) : ¼ ® 0 ® Im dn ® X ® X ® ¼ ,

¥ n-2 n-1 s²n(X ) : ¼ ® X ® X ® Ker dn ® 0 ®¼ ,

¥ n+1 n+2 t>n(X ) : ¼ ® 0 ® X ® X ® ¼ ,

¥ n-1 n t²n(X ) : ¼ ® X ® X ® 0 ®¼ .

For m ² n, we denote by K [m,n](B) the full subcategory of K (B) generated by complexes of the form: ¼ ® 0 ® X m®¼ ® X n-1® X n® 0 ®¼ , and denote by D [m,n](A) the full subcategory of D (A) generated by complexes of which homology Hi = 0 (i < m or n < i ).

2. Cotilting Bimodule Complexes and Morita Derived Duality

For a ring A , we denote by ModA (resp., A -Mod) the category of right (resp., left) A -modules, and denote by modA (resp., A -mod) the category of finitely presented right (resp., left) A -modules. We denote by InjA (resp.,A -Inj) the category of injective right (resp., left)

A -modules, and denote by PA (resp., AP) the category of finitely generated projective right (resp., left) modules. If A is a right coherent ring, then modA is an thick abelian subcategory

-, of ModA , and then D *(modA ) is equivalent to K *(PA ). Moreover, D *(modA ) is * equivalent to Dmod A (ModA), where * = - or b (see [4]).

For a right A- module UA over a ring A , we denote by add UA (resp., sum UA ) the category

of right A- modules which are direct summands of finite direct sums of copies of UA (resp.,

finite direct sums of copies of UA ).

¥ ¥ ¥ For a sequence {X i ; fi : X i ® X i+1 }i³1 of complexes in K (ModA ) (resp., D (ModA )), we have the following distinguished triangle in K (ModA ) (resp., D (ModA )):

3 1Ðshift ¥ ® ¥ ¥ ÅiXi ÅiXi ® X ®.

¥ ¥ We denote X by hlim X , and call it the homotopy colimit of the sequence [2]. i ®¥ i ¥ ¥ ¥ Similarly, for a sequence {X i ; fi : X i+1 ® X i }i³1 of complexes in K (ModA ) (resp., D (ModA )), we have the following distinguished triangle in K (ModA ) (resp., D (ModA )):

1Ðshift ¥ ¥ ® ¥ X ® PiXi PiXi ®.

¥ ¥ We denote X by hlim X , and call it the homotopy limit of the sequence. ¥¬i i According to [2], for a complex X ¥ Î K (ModA ), we have the following isomorphisms in D (ModA ):

¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ hlim X X hlim X X , hlim X X and hlim X X . n ®¥t³ -n @ , ¥ ¬ n s> -n @ n ®¥s² n @ ¥ ¬ n t² n @

Spaltenstein, Bšskstedt and Neeman defined the triangulated subcategory K s(InjA ) (resp., K s(ProjA )) of K (ModA ) which consists of special complexes of injective (resp., projective)

¥ right A- modules. Given a complex X Î D (ModA ), we have the isomorphism HomD (ModA)

¥ ¥ ¥ ¥ ¥ s (X , I ) @ HomK (ModA)(X , I ) for every complex I Î K (InjA ). Moreover, for every complex X ¥ Î D (ModA ), there exists a complex I ¥ Î K s(InjA ) which has a quasi-isomorphism X ¥ ® I ¥ in K (ModA). Similarly, given a complex X ¥ Î D (ModA ), we have the isomorphism

¥ ¥ ¥ ¥ ¥ s HomD (ModA)(P , X ) @ HomK (ModA)(P , X ) for every complex P Î K (InjA ). Moreover, for every complex X ¥ Î D (ModA ), there exists a complex P ¥ Î K s(ProjA ) which has a quasi-isomorphism P ¥ ® X ¥ in K (ModA ) (see [2], [12] for details).

¥ i i i+1 Let A and B be rings. A complex X = (X ; di : X ® X ) is called a B-A- bimodule

i complex provided that all X are B-A- bimodules and all di are B-A- bimodule morphisms.

Definitions. Let A be a right coherent ring and B a left coherent ring. A B - A -

bimodule complex BUA is called a cotilting B - A - bimodule complex provided that it satisfies the following:

4 ¥ b (C1) BUA is contained in Dmod A (ModA) as a right A- module complex , and is contained in b DB Ðmod(B- Mod) as a left B-module complex.

¥ b (C2r ) BUA belongs to K (InjA) as a right A- module complex;

¥ b (C2l ) BUA belongs to K (B-Inj) as a left B-module complex;

¥ ¥ (C3r ) HomD(ModA)(U ,U [i ]) = 0 for all i • 0;

¥ ¥ (C3l ) HomD(B-Mod)(U ,U [i ]) = 0 for all i • 0;

¥ ¥ (C4r ) the natural left multiplication morphism B ® HomD(ModA)(BUA , BUA ) is a ring isomorphism;

op ¥ ¥ (C4l ) the natural right multiplication morphism A ® HomD(B-Mod)(BUA , BUA ) is a ring isomorphism.

In case of B = A , we will call a cotilting A - A - bimodule complex a dualizing A- bimodule complex.

We say that A is a left Morita (resp., strong-Morita) derived dual of B if there exist contravariant continuous ¶-functors F : D (ModA ) ® D (B- Mod) and G : D (B- Mod) ® D (ModA ) which satisfy the condition (D1) (resp., the conditions (D1), (D2) and (D3)):

b b (D1) F and G induce a duality between Dmod A (ModA ) and DB Ðmod(B- Mod) ;

b (D2r ) the image of F|Db(ModA ) is contained in D (B- Mod) ;

b (D2l ) the image of G|Db(B-Mod) is contained in D (ModA ).

Remark. Let F : D (ModA ) ® D (B- Mod) and G : D (B- Mod) ® D (ModA) be ¶-functors satisfying that A is a left Morita derived dual of B. Then {F, G} is a right adjoint

b b pair as functors between Dmod A (ModA ) and DB Ðmod(B- Mod). It needs not be a right adjoint pair as functors between D (ModA ) and D (B- Mod), but we have the following statement.

Proposition 2.1. Let A be a right coherent ring, B a left coherent ring (resp., a left noetherian ring), and let F : D (ModA ) ® D (B- Mod) and GÕ : D (B- Mod) ® D (ModA) be contravariant continuous ¶-functors satisfying that A is a left Morita (resp., strong-Morita) derived dual of B. Then there exists a ¶-functor G : D (B- Mod) ® D (ModA) which 5 satisfies the following.

(a) {F, G} is a right adjoint pair as functors between D (ModA ) and D (B- Mod). (b) {F, G} induces that A is a left Morita (resp., strong-Morita) derived dual of B.

Proof. According to [9, Theorems 3.1 and 4.1], there exists a ¶-functor G : D (B- Mod)

® D (ModA) such that {F, G} is a right adjoint pair as functors between D (ModA ) and D

¥ b (B- Mod). By the above remark, for every complex Y in DB Ðmod(B- Mod), we have the following isomorphisms.

i ¥ ¥ ¥ H (GY ) @ HomD (ModA)(A,GY [i]) @ HomD(B-Mod)(Y ,FA[i]) ¥ i ¥ @ HomD(ModA)(A, GÕY [i]) @ H (GÕY ) for all i.

¥ b Then GY belongs to Dmod A (ModA ). Hence, by adjointness of F, G is isomorphic to GÕ as b b functors from DB Ðmod(B- Mod) to Dmod A (ModA ). Let l I(B) be the set of left ideals of B. In case of B being left Noetherian, there exists some n such that we have the following isomorphisms:

Hom ( Å B/J, FA[i]) @ P Hom (B/J , FA[i]) D(B-Mod) J ÎlI(B ) J ÎlI(B ) D(B-Mod) @ P Hom (A , GÕB/J[i]) J ÎlI(B ) D(ModA) @ Hom (A , P GÕB/J[i]) D(ModA) J ÎlI(B ) @ Hom (A , GÕ( Å B/J)[i]) D(ModA) J ÎlI(B ) = 0 for all i > n.

b ¥ b By Lemma 3.1 (a), we get FA Î D (B-Mod)fid . For every complex Y in D (B- Mod), there exist m ² n such that we have the following isomorphisms:

i ¥ ¥ H (GY ) @ HomD(ModA)(A,GY [i]) ¥ @ HomD(B-Mod)(Y ,FA[i]) = 0 for all i < m or i > n.

6 ¥ b b Then GY belongs to D (ModA), and therefore the image of G|Db(B-Mod) is contained in D (ModA). Hence F and G induce that A is a left strong-Morita derived dual of B

Lemma 2.2. Let A and B be rings, BUA a B - A - bimodule. Then {HomA(- , BUA) :

ModA ® B- Mod, HomB(- , BUA) : B- Mod ® ModA } is a right adjoint pair.

¥ ¥ ¥ Lemma 2.3. Let BUA be a B-A-bimodule complex. Then {RHom A(-, BUA ) : D (ModA )

¥ ¥ ® D (B-Mod), RHom B(-, BUA ) : D (B-Mod) ® D (ModA ) } is a right adjoint pair.

¥ ¥ Proof. According to [2], for a complex XA Î D (ModA ), there exist complex P Î

s ¥ ¥ ¥ K (ProjA ) such that X is isomorphic to P in D (ModA ). Similarly, for a complex BY Î D (B-Mod), there exist complex Q ¥ Î K s(B- Proj) such that Y ¥ is isomorphic to Q ¥ in D (B- Mod). Then we have the following isomorphisms:

¥ ¥ ¥ ¥ 0 ¥ ¥ ¥ ¥ ¥ HomD(ModA)(X , RHom B(Y , BUA )) @ H Hom A(P , Hom B(Q , BUA ))

0 ¥ ¥ ¥ ¥ ¥ @ H Hom B(Q , Hom A(P , BUA ))

¥ ¥ ¥ ¥ @ HomD(B-Mod)(Y , RHom A(X , BUA )).

Definition. Let U be a family of objects of D [m,n](ModA). We call a complex X ¥ in

¥ D (ModA ) a U-limit complex with ({Xi }i ³0 ; r ) if there exist an integer r and a sequence of the following distinguished triangles:

¥ ¥ ¥ U1 [-1]® X1 ® X0 ® ,

¥ ¥ ¥ U2 [-2]® X2 ® X1 ® , ¼

¥ ¥ ¥ Un [-n ]® Xn ® Xn-1 ® , ¼ ,

¥ ¥ ¥ ¥ where X and U belong to [r ] for all i ³ 1, such that X is isomorphic to hlim X in 0 i U ¥¬i i ¥ ¥ [m,n] D (ModA ). In case of U = addUA for some complex UA of D (ModA), we simply call a 7 ¥ U-limit complex a UA -limit complex.

Lemma 2.4. Let U be a family of objects of D [m,n](ModA). For a U-limit complex X ¥

¥ with ({Xi }i ³0 ; r ), the following hold.

¥ [s,t +k] (a) We have X k Î D (ModA) for all k ³ 0, where s = m - r and t = n - r .

¥ ¥ (b) We have an isomorphism s²s+k-2X k @ s²s+k-2X k-1 in D (ModA ) for every k ³ 1, where s = m - r . [ m,n] (b) If A is a right coherent ring, and if U is a family of objects of D modA (ModA), then

¥ + X belongs to D mod A (ModA ).

Proof. It is straightforward.

¥ Lemma 2.5. Let BUA be a B-A-bimodule complex satisfying the conditions (C1) and (C2r), and U a family of complexes in D [m,n](ModA ). If X ¥is a U-limit complex with a

¥ ¥ ¥ ¥ sequence {X } , then the induced natural morphism hlim Hom (X , U ) ® i i ³0 i ®¥ A i B A ¥ ¥ ¥ Hom (hlim X , U ) is an isomorphism in D (B-Mod). A ¥¬i i B A

Proof. It is easy to see that we have the following commutative diagram in D (B-Mod):

¥ ¥ ¥ ¥ ¥ ¥ Hom A(Xi , BUA ) = Hom A(Xi , BUA ) ¯ ¯

¥ ¥ ¥ h ¥ ¥ ¥ hlim Hom (X , U ) ¾®¾ Hom (hlim X , U ). i ®¥ A i B A A ¥¬i i B A

¥ [0,t] We may assume BUA is contained in K (InjA ). Given an integer k , we have the following isomorphisms:

k ¥ ¥ ¥ ¥ ¥ H Hom (hlim X , U ) @ Hom (hlim X , U [k ]) A ¥¬i i B A D(ModA) ¥¬i i B A ¥ ¥ @ Hom (s hlim X , U [k ]) D(ModA) ²t -k¥¬i i B A ¥ ¥ @ HomD(ModA)(s²t -kXp , BUA [k ]) for some p >> 0

¥ ¥ @ HomD(ModA)(Xp , BUA [k ]) 8 k ¥ ¥ ¥ @ H Hom A(Xp , BUA ).

Moreover, there exists an integer q such that we have the following isomorphisms for all j ³ 0:

k ¥ ¥ ¥ ¥ ¥ H Hom A(Xq , BUA ) @ HomD(ModA)(Xq , BUA [k ])

¥ ¥ @ HomD(ModA)(s²t -kXq , BUA [k ])

¥ ¥ @ HomD(ModA)(s²t -kXq+j , BUA [k ])

k ¥ ¥ ¥ @ H Hom A(Xq+j , BUA ).

k ¥ ¥ ¥ k ¥ ¥ ¥ Then we have the isomorphism H Hom (X , U ) @ H hlim Hom (X , U ). For all integers A q B A i ®¥ A i B A r ³ max(p, q ), we have the following commutative diagram:

k ¥ ¥ ¥ k ¥ ¥ ¥ H Hom A(Xr , BUA ) = H Hom A(Xr , BUA ) ¯ ¯

k ¥ ¥ ¥ H kh k ¥ ¥ ¥ H hlim Hom (X , U ) ¾®¾ H Hom (hlim X , U ), i ®¥ A i B A A ¥¬i i B A

where vertical arrows are isomorphisms. Therefore Hkh is an isomorphism, and hence h is an isomorphism in D (B-Mod).

¥ Theorem 2.6. Let A be a right coherent ring, B a left coherent ring, BUA a B-A-bimodule Ð complex satisfying the conditions (C1), (C2r ), (C3r ) and (C4r ). Then {D B mod (B- Mod);

¥ ¥ - ¥ ¥ + Hom A(-, BUA ), R Hom B(-, BUA )} is a localization duality of D mod A (ModA ), and the image

¥ b of Hom (-, U )| b is contained in D (B- Mod). Moreover, every complex in A B A D modA(ModA ) B mod + ¥ ¥ ¥ - ¥ ¥ D mod A (ModA ) is a UA -limit complex if and only if Hom A(-, BUA ) and R Hom B(-, BUA ) Ð + induce the duality between D B mod (B- Mod) and D mod A (ModA ).

+ ¥ ¥ ¥ ¥ Proof. The condition (C2r ) implies the existence of R Hom A(-, BUA ) @ Hom A(-, BUA ):

+ - ¥ ¥ ¥ D (ModA ) ® D (B-Mod). We have Hom A(PA, BUA ) belongs to addBU for all P Î PA .

+ ¥ ¥ Then, according to [4, Chapter I, Proposition 7.3], we can consider R Hom A(-, BUA ) :

+ Ð + ¥ D (ModA) ® D (B- Mod), and the image of R Hom (-, U )| b is contained modA B mod A B A D modA(ModA ) 9 b - ¥ ¥ Ð + in D B mod (B- Mod). It is clear that R Hom B(-, BUA ) : D B mod (B- Mod) ® D (ModA ) Ð - Ð exists. Since D B mod (B- Mod) is equivalent to D (B- mod), D B mod (B- Mod) is equivalent - ¥ Ð ¥ - to K (BP). Given a complex X Î D B mod (B- Mod), there exists a complex P Î K (BP) ¥ ¥ Ð - ¥ ¥ ¥ such that X is isomorphic to P in D B mod (B- Mod). Since R Hom B(P , BUA ) is isomorphic

¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ to R -Hom (hlim P , U ) hlim R -Hom ( P , U ), R -Hom (P , U ) is a U -limit B n ®¥t³-n B A @ ¥¬n B t³-n B A B B A A - ¥ ¥ ¥ + complex. By Lemma 2.4, R Hom B(P , BUA ) is contained in D mod A (ModA ). Also, the

¥ + ¥ - ¥ ¥ conditions (C3r) and (C4r ) imply that the natural morphism t³-nP ® R Hom A(R Hom B(t³-nP ,

¥ ¥ BUA ), BUA ) is an isomorphism in D (B- Mod). Therefore, according to Lemma 2.5, we have the following commutative diagram in D (B- Mod):

¥ + ¥ ¥ ¥ ¥ ¥ hlim P hlim R Hom (R -Hom ( P , U ), U ) n ®¥t³-n ® n ®¥ A B t³-n B A B A ¯

+ ¥ ¥ ¥ ¥ ¥ R Hom ( hlim R -Hom ( P , U ), U ) A ¥¬n B t³-n B A B A ¯

¥ + ¥ ¥ ¥ ¥ ¥ hlim P R Hom (R -Hom (hlim P , U ), U ) n ®¥t³-n ® A B n ®¥t³-n B A B A ¯¯

¥ + ¥ - ¥ ¥ ¥ ¥ P ® R Hom A(R Hom B(P , BUA ), BUA ),

¥ + ¥ - ¥ ¥ where vertical arrows are isomorphisms in D (B- Mod). Hence P ®R Hom A(R Hom B(P ,

¥ ¥ BUA ), BUA ) is an isomorphism in D (B- Mod).

¥ ¥ ¥ ¥ By the above, it is easy to see if Hom A(-, BUA ) and Hom B(-, BUA ) induce the duality between Ð + + ¥ D B mod (B- Mod) and D mod A (ModA ), then every complex in D mod A (ModA ) is a UA -limit

¥ + ¥ complex. Conversely, if every complex X in D mod A (ModA ) is a UA -limit complex, then there exist an integer r and a sequence of the following distinguished triangles:

¥ ¥ ¥ U1 [-1]® X1 ® X0 ® ,

¥ ¥ ¥ U2 [-2]® X2 ® X1 ® , ¼

¥ ¥ ¥ Un [-n ]® Xn ® Xn-1 ® , ¼ , 10 ¥ ¥ ¥ ¥ ¥ where X and U belong to (addU )[r ] for all i ³ 1, such that X is isomorphic to hlim X 0 i A ¥¬i i ¥ ¥ ¥ ¥ ¥ ¥ in D (ModA). Since Ui [-i ]® Hom B(Hom A(Ui [-i ], BUA ), BUA ) is an isomorphisim in D

¥ ¥ ¥ ¥ ¥ ¥ (ModA) for all i, the natural morphism Xi ® Hom B(Hom A(Xi , BUA ), BUA ) is an isomorphism

¥ ¥ ¥ in D (ModA) for all i. By Lemma 2.5, the natural morphism hlim X ® Hom (Hom (hlim ¥¬i i B A ¥¬i ¥ ¥ ¥ ¥ ¥ Ð Xi , BUA ), BUA ) is an isomorphism in D (ModA). Therefore, Hom B(-, BUA ): D B mod (B- Mod)

+ ¥ ¥ ¥ ¥ ® D mod A (ModA ) is dense, and hence Hom A(-, BUA ) and Hom B(-, BUA ) induce the duality Ð + between D B mod (B- Mod) and D mod A (ModA ).

¥ Corollary 2.7. Let A be a right coherent ring, B a left coherent ring, BUA a B-A-bimodule b complex satisfying the conditions (C1), (C2r ), (C2l ), (C3r ) and (C4r ). Then {D B mod (B- ¥ ¥ ¥ ¥ b Mod); Hom A(-, BUA ), Hom B(-, BUA )} is a localization duality of D modA (ModA ).

b ¥ ¥ Proof. By the condition (C2l ), it easy to see that the image of R Hom B(-, BUA ) @ ¥ ¥ b Hom B(-, BUA ) is contained in D modA (ModA ). We are done by Theorem 2.6.

¥ Corollary 2.8. Let A be a right coherent ring, B a left coherent ring, BUA a cotilting B-A-bimodule complex . Then A is a left strong-Morita derived dual of B , and there is a duality between DmodA(ModA ) and DB-mod(B- Mod).

¥ ¥ ¥ ¥ Proof. It is clear that Hom A(-, BUA ) and Hom B(-, BUA ) are continuous ¶-functors. According to Lemma 2.3 and Corollary 2.7, A is a left strong-Morita derived dual of B .

¥ ¥ ¥ ¥ Since Hom A(-, BUA ) and Hom B(-, BUA ) are way-out in both directions, by [4, Chapter I, Proposition 7.1], we deduce the assertion.

Let A be an abelian category, B a full subcategory of A. We call an object X Î A a finitely embedding cogenerator for B provided that every object in B has an injection to some finite direct sum of copies of X in A.

¥ Theorem 2.9. Let A be a right coherent ring , B a left coherent ring, and BUA a B-A-bimodule complex which satisfies the conditions (C1) and (C2r ). Assume that the

11 ¥ ¥ Ð + ¥ image of Hom A(-, BUA ) : D modA (ModA ) ® D B Ðmod (B- Mod) contains B- mod. If E is a

-s 0 + ¥ complex E ® ¼ ® E ® ¼ in K (B- Inj) which is isomorphic to BUA in D (B-Mod), then

Å E i is a finitely embedding cogenerator for B- mod, and then P E i is a finitely embedding i ³ s i ³ s injective cogenerator for B- mod.

Ð - Ð Proof. Since D modA (ModA ) is equivalent to D (modA ), D modA (ModA ) is equivalent

- ¥ to K (sumAA). By assumption, for every X Î B- mod, there exists a complex P in

- ¥ ¥ ¥ b ¥ ¥ K (sumAA) such that Hom A(P , BUA ) is isomorphic to X in D (B- Mod). Since Hom A(P ,

¥ ¥ i ¥ ¥ BUA ) is BU -limit complex, there exists an integer n such that we have isomorphism H Hom A(P ,

¥ i ¥ ¥ ¥ ¥ n BUA ) @ H Hom A(t³nP , BUA ) for all i ² 0. We may assume t³nP is a complex P ® ¼ ®

m i i ¥ ¥ ¥ P , where P Î sumAA (n ² i ² m ). Then HomA(P , BUA ) is isomorphic to Ei for some Ei Î

¥ ¥ ¥ ¥ sumE (n ² i ² m ). Therefore Hom A(t³nP , BUA ) is isomorphic to a iterated mapping cone

¥ ¥ ¥ ¥ ¥ ¥ complex En [n ]ÅEn+1 [n +1] Å ¼ ÅEm [m ]. The complex En [n ]ÅEn+1 [n +1] Å ¼ ÅEm [m ] is of the form I -m-s® ¼ ® I -1 ® I 0 ® ¼ , where I j Î add( Å E i). Then we have the i ³ s following exact sequences:

-m-s -1 0 ® I ® ¼ ® I ® Imd-1 ® 0 ...... (1),

0 ® Imd-1 ® Kerd0 ® X ® 0 ...... (2).

i Since I is injective (-s ² i ² -1), Imd-1 is injective. Therefore, the exact sequence (2) splits,

and hence X has an injection to I 0. By I 0 Î add( Å E i ), X is embedded in some finite i ³ s direct sum of copies of Å E n. i ³ s

¥ Corollary 2.10. Let A be a right coherent ring, B a left coherent ring, BUA a B-A-bimodule

0 n complex BUA ® ¼ ® BUA satisfying the conditions (C1), (C2r ), (C2l ), (C3r ) and (C4r ). n Then Å U i is a finitely embedding injective cogenerator for B- mod. i 0

Proof. By Corollary 2.7 and Theorem 2.9.

¥ Corollary 2.11. Let A be a right coherent ring, B a left coherent ring, BUA a cotilting 12 n B-A-bimodule complex U 0® ¼ ® U n . Then Å U i is a finitely embedding injective B A B A i 0 cogenerator for B- mod.

¥ Corollary 2.12. Let A be a right coherent ring, B a left Noetherian ring, BUA a cotilting

0 n B-A-bimodule complex BUA ® ¼ ® BUA . Then every injective indecomposable left

i B-module is isomorphic to a direct summand of some BUA .

¥ Lemma 2.13. Let A , B, C and D be rings, AX B a bounded above A-B-bimodule

- ¥ complex which is contained in K (PB), CYB a bounded below C-B-bimodule complex, and

¥ CZD a bounded C-D-bimodule complex . Then we have the natural A-D-bimodule complex

¥ · ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ isomorphism AX B ÄBHom C(CYB , CZD ) @ Hom C(Hom B(AX B , CYB ), CZD ).

Proof. Let X be a A-B- bimodule which is finitely generated projective as a right B- module, Y a C-B- bimodule, and Z a C-D-bimodule. Then we have the natural A-D-

bimodule isomorphism X ÄBHomC(Y, Z ) ® HomC(HomB(X , Y ), Z ) by elementary correspondence (x Äf a (g a f (g (x )))). Then we clearly get the statement.

¥ Following Rickard [11], we call an A-B- bimodule complex ATB a tilting A-B-bimodule complex if it satisfies the conditions (C3r ), (C3l ), (C4r ), (C4l ) and

¥ b b (T1) ATB belongs to K (PB) as a right B- module complex, and belongs to K (AP) as a left A-module complex.

In case of finite dimensional k- algebras over a k , we defined a cotilting module

complex by using a duality Homk(-,k ) : modA ® A- mod [7]. We construct a cotilting bimodule complex by using dualizing complexes.

Theorem 2.14. Let R be a commutative Noetherian ring with a dualizing complex w ¥, A

¥ and B R-algebras which are finitely generated R-modules. If ATB is a tilting A-B- bimodule

¥ ¥ ¥ complex, then Hom R(ATB , w ) is a cotilting B-A-bimodule complex.

13 ¥ ¥ ¥ b Proof. It is clear that Hom R(ATB , w ) is contained in K (InjA) as a right A- module

b ¥ ¥ ¥ complex, and is contained in K (B-Inj) as a left B-module complex. Since Hom R(ATB , w ) is

¥ ¥ ¥ ¥ ¥ ¥ ¥ an w -limit complex with a sequence {Hom R(t³-n ATB , w )}, Hom R(ATB ,w ) is contained in b D modR (ModR ). Since A and B are finitely generated R- modules, every homology of

¥ ¥ ¥ Hom R(ATB , w ) is a finitely generated R- module, and hence finitely generated as a right A-

¥ ¥ ¥ b module and as a left B- module. Therefore, Hom R(ATB , w ) is contained in Dmod A (ModA) b as a right A- module complex, and is contained in D B mod (B- Mod) as a left B-module

¥ ¥ ¥ complex. In order that Hom R(ATB , w ) satisfies the conditions (C3r ) and (C4r ), it suffices

b ¥ b ¥ ¥ ¥ ¥ ¥ ¥ to show that the natural morphism AA ® R Hom B(R Hom A(AA, Hom R(ATB , w )), Hom R(ATB ,

¥ b w )) is an isomorphism in Dmod A (ModA). By Lemma 2.13, we have the following isomorphisms in D (ModA ):

b ¥ b ¥ ¥ ¥ ¥ ¥ ¥ ¥ R Hom B(R Hom A(AA , Hom R(ATB , w )), Hom R(ATB , w ))

¥ ¥ ¥ ¥ ¥ ¥ ¥ @ Hom B(Hom R(TB , w ), Hom R(ATB , w ))

¥ ¥ · ¥ ¥ ¥ ¥ @ Hom R(ATB ÄBHom R(TB , w ), w )

¥ ¥ ¥ ¥ ¥ ¥ ¥ @ Hom R(Hom R(Hom B(ATB , TB ), w ), w ).

¥ ¥ ¥ ¥ Since ATB is a tilting A-B- bimodule complex, the natural morphism AA ® Hom B(ATB , TB ) is a quasi-isomorphism in K (ModA ). By the duality of w ¥, we have the following isomorphisms:

¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ Hom R(Hom R(Hom B(ATB , TB ), w ), w ) @ Hom R(Hom R(AA, w ), w )

@ AA .

¥ ¥ ¥ Similarly, Hom R(ATB , w ) satisfies the conditions (C3l ), (C4l ) .

We get the non-commutative ring version of results of Grothendieck and Hartshorne [4, Chapter V, Proposition 2.4].

Corollary 2.15. Let R be a commutative Noetherian ring, A an R-algebra which is finitely generated as an R-module. If w ¥ is a dualizing R-module complex, then

14 ¥ HomR(A , w ) is a dualizing A-bimodule complex.

3. A Morita Duality Theorem for Derived Categories

Let k be a commutative ring. We call an k- algebra A a projective k- algebra if A is projective as a k- module. Let A, B and C be projective k- algebras. According to [3], a

op projective (resp., injective) B ÄkA- module is projective (resp., injective) as both a right A- module and a left B- module. According to [11], [13] and [2], we have the following derived functors:

¥ op op op op R Hom A(-,-) : D (ModB ÄkA ) ´D (ModC ÄkA )®D (ModC ÄkB ),

· L op op op - ÄA - : D (ModB ÄkA )´D (ModA ÄkC )®D (ModB ÄkC ).

b b Let D (ModA )fid be the triangulated subcategory of D (ModA ) generated by complexes which are isomorphic to complexes in K b(InjA ).

Lemma 3.1. Let A be a ring, and rI(A ) the set of right ideals of A . For a complex X ¥ Î D b(ModA ), the following hold.

¥ (a) If there exist an integer n such that Hom b ( Å A/I, X [i ]) = 0 for all i > n , D (ModA ) I ÎrI(A ) ¥ b then X belongs to D (ModA )fid . (b) In case of A being a right Artinian ring, if there exist an integer n such that

¥ ¥ b HomDb(ModA )(A/radA , X [i ]) = 0 for all i > n , then X belongs to D (ModA )fid .

Proof. (a) By Baer condition. (b) By [1].

Lemma 3.2. Let A be a right coherent projective k-algebra, B a left coherent projective

¥ b k-algebra. Let BVA be a B-A-bimodule complex which belongs to D (ModA )fid as a right

b A-module complex, and belongs to D (B- Mod)fid as a left B-module complex. Then there

¥ b exists a bounded B-A-bimodule complex BUA , which belongs to K (InjA ) as a right A-module

b ¥ complex, and belongs to K (B- Inj) as a left B-module complex, such that BUA is isomorphic 15 ¥ op to BVA in D (ModB ÄkA ).

Proof. See [13, Proposition 2.4].

By the above lemma, we can replace the conditions (C2r ) and (C2l ) of cotilting bimodule complexes by the following conditions:

¥ b (C2 r ) BUA belongs to D (ModA )fid as a right A- module complex;

¥ b (C2 l ) BUA belongs to D (B- Mod)fid as a left B-module complex.

Theorem 3.3. Let A be a right coherent projective k-algebra and B a left Noetherian projective k-algebra. The following are equivalent.

(a) A is a left strong-Morita derived dual of B.

¥ (b) There exists a cotilting B-A-bimodule complex BUA .

Proof. (b) Þ (a): By Corollary 2.8. (a) Þ (b): Let F : D (ModA ) ® D (B- Mod) and GÕ : D (B- Mod) ® D (ModA) be continuous ¶-functors satisfying that A is a left srong-Morita derived dual of B. By

Proposition 2.1, we can take a right adjoint pair {F : D (ModA ) ® D (B- Mod), G : D (B- Mod) ® D (ModA )} satisfying that A is a left strong-Morita derived dual of B. Let X ¥ be a b complex GB Î D modA (ModA ). Then we have the following isomorphisms:

¥ ¥ HomD (ModA )(X , X [i ]) = HomD (ModA ) (GB , GB [i ])

@ HomD (B-Mod)(B , B [i ]) = 0 for all i • 0.

¥ - op According to [5], there exists a B-A- bimodule complex BUA Î K (ProjB ÄkA ) such that

¥ ¥ BUA is isomorphic to XA in D (ModA ), and that the natural left multiplication morphism B

¥ ¥ ¥ ® HomD(ModA)(BUA , BUA ) is a ring isomorphism. Then BUA satisfies the conditions (C3r ) and

¥ (C4r ). Since B @ EndD(ModA)(BUA ) @ EndD(ModA)(GB), we have the following isomorphisms as B-A- bimodules:

16 i ¥ ¥ H (BUA ) @ HomD (ModA ) (A, BUA [i ])

@ HomD (ModA )(A, GB[i ])

@ HomD (B-Mod)(B, FA [i ]).

b ¥ b Since FA belongs to D B mod (B- Mod), then BUA belongs to D B mod (B- Mod). Therefore

U ¥ satisfies the condition (C1). Since Å A/I belongs to D b(ModA ), F( Å A/I ) B A I ÎrI(A ) I ÎrI(A ) belongs to D b(B- Mod). Then there exists an integer n such that we have

Hom ( Å A/I , U ¥[i ]) @ Hom ( Å A/I , GB [i ]) D (ModA ) I ÎrI(A ) B A D (ModA ) I ÎrI(A ) @ Hom (B, F( Å A/I ) [i ]) D (B-Mod) I ÎrI(A ) = 0 for all i > n.

¥ b By Lemma 3.1 (a), we get BUA Î D modA (ModA )fid . According to Lemma 3.5, for every

¥ - ¥ ¥ ¥ ¥ complex P Î K (BP), we have an isomorphism GP @ R Hom B(P , BUA ) in D (ModA ). Since B is left Noetherian, by the continuity of G, we have

R Hom¥ ( Å B/J , U ¥) @ P R Hom¥ (B/J , U ¥) B J ÎlI(B ) B A J ÎlI(B ) B B A @ P G(B/J ) J ÎlI(B ) @ G( Å B/J ). J ÎlI(B )

Then R Hom¥ ( Å B/J , U ¥) belongs to D b(ModA ). Since {R Hom¥ (-, U ¥) : B J ÎlI(B ) B A A B A ¥ ¥ D (ModA) ® D (B-Mod), R Hom B(-, BUA ) : D (B-Mod) ® D (ModA )} is a right adjoint pair, there exists an integer n such that we have

Hom ( Å B/J , U ¥[i ]) @ Hom ( Å B/J , R Hom¥ (A , U ¥)[i ]) D (B- Mod) J ÎlI(B ) B A D (B- Mod) J ÎlI(B ) A A B A @ Hom (A , R Hom¥ ( Å B/J , U ¥)[i ]) D (ModA) A B J ÎlI(B ) B A = 0 for all i > n .

¥ b By Lemma 3.1 (a), we get BUA Î D B mod (B- Mod)fid . Since the natural morphism B ®

¥ ¥ ¥ b op R Hom A(BUA , BUA ) is an isomorphism in D (ModB ÄkB ), we have an isomorphism P ® 17 ¥ ¥ ¥ R Hom B(R HomA(P, BUA ), BUA ), for every finitely generated projective left B- module. Then

¥ ¥ ¥ ¥ ¥ ¥ ¥ b we have an isomorphism P ® R Hom A(R Hom B(P , BUA ), BUA ) for every P Î K (BP). By

¥ - ¥ the duality, there exists a complex Q Î K (BP) such that AA @ GQ in D (ModA ). According

¥ ¥ ¥ ¥ to Lemma 3.5, we get an isomorphism GQ @ R Hom B(Q , BUA ) in D (ModA ). Since

¥ ¥ ¥ ¥ ¥ ¥ ¥ R Hom B(Q , BUA ) is a UA -limit complex with a sequence {R Hom B(t³-nQ , BUA )}, we have the following isomorphisms in D (B- Mod):

¥ ¥ ¥ ¥ ¥ ¥ hlim Q hlim R Hom (R Hom ( Q , U ), U ) n ®¥t³-n ® n ®¥ A B t³-n B A B A ¯

¥ ¥ ¥ ¥ ¥ R Hom ( hlim R Hom ( Q , U ), U ) A ¥¬n B t³-n B A B A ¯

¥ ¥ ¥ ¥ ¥ ¥ hlim Q R Hom (R Hom (hlim Q , U ), U ) n ®¥t³-n ® A B n ®¥t³-n B A B A ¯¯

¥ ¥ ¥ ¥ ¥ ¥ Q ® R Hom A(R Hom B(Q , BUA ), BUA ), where the vertical arrows are isomorphisms in D (B- Mod). Then, by the dualities and the

¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ property of right adjoint pairs, R Hom B(Q , BUA ) ®R Hom B(R Hom A(R Hom B(Q , BUA ),

¥ ¥ ¥ ¥ ¥ BUA ), BUA ) is an isomorphism in D (ModA ), and hence AA ® R Hom B(R Hom A(AA, BUA ),

¥ ¥ BUA ) is an isomorphism in D (ModA ). This implies that BUA satisfies the conditions (C3l ) and (C4l ).

¥ b Lemma 3.4. Let UA be a complex in D (ModA ) which satisfies the condition (C3r ),

¥ ¥ ¥ ¥ ¥ and X a UA -limit complex with ({Xi }i ³0 ; 0). Then we have HomD (ModA )(Xk , UA [l ]) = 0 for all l < - k .

Lemma 3.5. In the situation of the proof in Theorem 3.3, for every complex P ¥ Î

- ¥ ¥ ¥ ¥ K (BP), we have an isomorphism GP @ R Hom B(P , BUA ) in D (ModA ).

¥ ¥ ¥ -1 0 Proof. Let H := R Hom B(-, BUA ), P a complex ¼® P ® P ® 0 ®¼ which belongs

- ¥ ¥ to K (BP), and Pi := t³-iP . Then we have the following sequences of distinguished triangles: 18 -1 ¥ 0 -1 ¥ 0 GP [-1] ® GP1 ® GP ®, HP [-1] ® HP1 ® HP ®,

-2 ¥ ¥ -2 ¥ ¥ GP [-2] ® GP2 ® GP1 ®, HP [-2] ® HP2 ® HP1 ®, ¼ ¼

-i ¥ ¥ -i ¥ ¥ GP [-i ] ® GPi ® GPi-1 ® , HP [-i ] ® HPi ® HPi-1 ® , ¼ , ¼ .

By inductive step, we construct isomorphisms between distinguished triangles:

-i zi ¥ yi ¥ xi -i GP [-i ] ¾®¾ GPi ¾®¾ GPi-1 ¾®¾ GP [-i +1]

¯bi ¯ai ¯ai-1 ¯bi[1]

-i wi ¥ vi ¥ ui -i HP [-i ] ¾®¾ HPi ¾®¾ HPi-1 ¾®¾ HP [-i +1],

¥ 0 ¥ where P0 := P . Since the natural morphisms B @ HomD (B-Mod)(GB , GB ) @ HomD (B-Mod)( BUA ,

¥ ¥ ¥ BUA ) are isomorphisms, the isomorphism GB @ R Hom B(B, BUA ) induces the isomorphism

¥ ¥ ¥ ¥ -i HomD (B-Mod)(GB , GB ) @ HomD (B-Mod)(R Hom B(B, BUA ), R Hom B(B, BUA )). Since all GP

-i ¥ and all HP belong to addUA , we can choose isomorphisms a0 and b1 , and therefore we can

choose an isomorphism a1 . Assume we have isomorphisms ai-1 , ai and bi which satisfy the

-i-1 -i-1 above condition. We also can choose an isomorphism bi+1: GP [-i -1] ® HP [-i -1] such

that bi+1[1]oxi+1 ozi = ui+1 owi obi . Since (bi+1[1]oxi+1 - ui+1 owi )ozi = bi+1[1]oxi+1 ozi - ui+1 owi obi =

¥ -i-1 0, by the property of distinguished triangles, there exists a morphism s : GPi-1 ® HP [-i ]

¥ -i-1 such that s oyi = bi+1[1]oxi+1 - ui+1 owi . But, by Lemma 3.4, HomD(ModA)(GPi-1 , HP [-i ]) = 0.

¥ ¥ Therefore bi+1[1]oxi+1 = ui+1 owi , and hence we also can choose ai+1 : GPi+1 ® HPi+1 which satisfies the above condition. Since G and H are contravariant continuous ¶-functors, we have the following isomorphisms in D (ModA ):

¥ ¥ ¥ ¥ ¥ ¥ GP @ Ghlim P @ hlim GP @ hlim HP @ Hhlim P @ HP . i ®¥ i ¥¬i i ¥¬i i i ®¥ i

Remarks. The conditions (D2r ) and (D2l ) are closely related to the property of finite injective dimension of complexes. Indeed, let R be a commutative Noetherian regular ring

19 of infinite Krull dimension, and A := R [X]/(X2 - a ), where a is a non-zero element in N 2 for some maximal N of R . Then A is a commutative locally Gorenstein ring of infinite Krull dimension which is non-regular. The bimodule A is a pointwise dualizing complex, b b but is not a dualizing complex. Moreover, A induces a duality D modA (ModA ) ® D modA (ModA ) (oral communication with Y. Yoshino). I donÕt know if an arbitrary locally Gorenstein b ring A induces a self-duality on D modA (ModA ), or equivalently if for every prime ideal P of i an arbitrary locally Gorenstein ring A, there is some integer n such that ExtA(A/P,A) = 0 for all i > n. In case of Artinian rings, we can delete the conditions (D2r ) and (D2l ).

Corollary 3.6. Let A be a right Artinian projective k-algebra, B a left Artinian projective k- algebra. Then the following are equivalent.

(a) A is a left Morita derived dual of B. (b) A is a left strong-Morita derived dual of B.

¥ (c) There exists a cotilting B-A-bimodule complex BUA .

Proof. By Theorem 3.3, it remains to show that (a) implies (c). By the proof of Theorem

¥ b b 3.3, it suffices to show that BUA belongs to D modA (ModA )fid and D B mod (B- Mod)fid . Since A is right Artinian and B is left Artinian, in the proof of Theorem 3.3, we can replace

Å A/I and Å B/J by A /radA and B /radB , respectively. We are done by Lemma I ÎrI(A ) J ÎlI(B ) 3.1 (b).

We get a non-commutative ring version of results of Grothendieck and Hartshone [4, Chapter V, Theorem 3.1] or Yekutieli [13, Theorem 3.9].

Proposition 3.7. Let A be a local right coherent projective k-algebra, B a left coherent

¥ ¥ projective k-algebra, and BUA a cotilting B-A-bimodule complex . Let BVA be any B-A-bimodule

+ op ¥ complex in D (ModB ÄkA ). Then BVA is a cotilting B-A-bimodule complex if and only if

¥ there exist an invertible A-bimodule L and some integer n such that BVA is isomorphic to

¥ + op BUA ÄAL [n ] in D (ModB ÄkA ).

20 ¥ ¥ Proof. Let L be an invertible A- bimodule. For an integer n, let BVA := BUA ÄAL [n ].

¥ By adjointness and Lemma 2.2 concerning L, it is not difficult to see that BVA satisfies the

¥ conditions of a cotilting bimodule complex. Conversely, let BVA be a cotilting B-A- bimodule

¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ complex. Then Hom B(Hom A(-, BUA ), BVA ) and Hom B(Hom A(-, BVA ), BUA ) : b b b D modA (ModA ) ® D modA (ModA ) are derived equivalences. Since D modA (ModA ) @

b -,b D (modA ) @ K (PA), by Lemma 2.13, we have the following isomorphisms:

¥ ¥ ¥ ¥ · L ¥ ¥ ¥ Hom A(Hom B(-, BUA ), BVA ) @ -ÄA Hom B(BUA , BVA ),

¥ ¥ ¥ ¥ · L ¥ ¥ ¥ Hom A(Hom B(-, BVA ), BUA ) @ -ÄA Hom B(BVA , BUA ).

¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ Let M and N be A- bimodule complexes Hom B(BUA , BVA ) and Hom B(BVA , BUA ), respectively. ¥ ¥ b It is clear that M and N are contained in D modA (ModA ). By the dualities, we have the following isomorphisms in D (A- Mod):

¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ HomD (B-Mod)(BUA , BV [i ]) @ HomD (ModA )(Hom B(BV , BUA ), Hom B(BUA , BUA )[i ])

¥ ¥ ¥ @ HomD (ModA )(Hom B(BV , BUA ), AAA [i ])

i ¥ ¥ @ HR Hom A(NA , AAA ) for all i .

¥ b ¥ b Then M belongs to D A mod (A- Mod). Similarly, N belongs to D A mod (A- Mod). Also,

¥ · L ¥ ¥ · L ¥ op M ÄA N and N ÄA M are isomorphic to A in D (ModA ÄkA ). Let p be the largest integer such that Hp(M ¥) • 0, and let q be the largest integer such that Hq(N ¥) • 0. Then we

p ¥ q ¥ p+q ¥ · L ¥ q ¥ p ¥ p+q ¥ · L ¥ have H (M )ÄAH (N ) @ H (M ÄA N ) and H (N )ÄAH (M ) @ H (N ÄA M ). Let X :=

p ¥ q ¥ H (M ) and Y := H (N ). We consider the surjection X ÄAY ® X /X (radA ) ÄAY /(radA )Y . Since X and Y are finitely generated A- modules on both sides, X /X (radA ) and Y /(radA )Y

p ¥ q ¥ are nonzero. By locality of A , X /X (radA ) ÄAY /(radA )Y is non-zero, and H (M )ÄAH (N )

q ¥ p ¥ p ¥ is non-zero. Similarly, H (N )ÄAH (M ) is non-zero. Then p +q = 0 and H (M ) is an invertible A- bimodule with inverse Hq(N ¥). Let H-q(M ¥) and Hq (N ¥) be L and L*, respectively.

By projectivity of L and L*, we have M ¥ @ MÕ ¥ÅL [q ] in D (ModA ) and N ¥ @ NÕ ¥ÅL* [-q ] in D (A-Mod). Then we have the following isomorphisms in D (Modk ):

21

¥ · L ¥ A @ M ÄA N

· L · L ¥ ¥ · L ¥ · L ¥ @ L [q ]ÄA L* [-q ]ÅL [q ]ÄA NÕ ÅMÕ ÄA L* [-q ]ÅMÕ ÄA NÕ .

· L ¥ ¥ · L ¥ · L ¥ ¥ ¥ Then L [q ]ÄA NÕ ÅMÕ ÄA L* [-q ]ÅMÕ ÄA NÕ is acyclic, and MÕ and NÕ are acyclic.

¥ ¥ op Therefore M and NÕ are isomorphic to L [q ] and L* [-q ] in D(ModA ÄkA ), respectively.

op Hence we have the following isomorphisms in D (ModB ÄkA ):

¥ ¥ ¥ ¥ ¥ ¥ BVA @ Hom A(Hom B(BUA , BUA ), BVA )

¥ · L ¥ ¥ ¥ @ BUA ÄA Hom B(BUA , BVA )

¥ · L @ BUA ÄA L [q ] .

Remark. In Proposition 3.7, we can replace "cotilting bimodule complex" by "tilting bimodule complex" under the condition that A is a local projective k-algebra and that B is a projective k-algebra.

Example. For the uniqueness of the cotilting bimodule complex, we need the condition that A is a local ring. Indeed, let A be a finite dimensional k- algebra over a field k which has the following quiver with relations:

¾®¾a 12· · , ¬¾¾b

with aba = bab = 0. Then A , A e1Äke1A ® A and A e2Äke2A ® A are dualizing A- bimodule complexes, where morphisms are natural multiplications.

References

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153 (1992), 41-84.

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