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 ring 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 module

categories [10]. He also introduced tilting bimodule complexes in case of projective k-

algebras over a commutative ring 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

1

rings, that is, the relations between cotilting bimodule complexes and dualities for derived

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,

F} is a right adjoint pair, and that the natural morphism idV ® GoF 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

2

of an abelian category A, let K *,b(B) be the full subcategory of K *(B) generated by

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).

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

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

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

t>n(X ¥) : ¼ ® 0 ® X n+1® X n+2® ¼ ,

t²n(X ¥) : ¼ ® X n-1® X n® 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 DmodA* (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

ÅiXi¥ ®1Ðshift ÅiXi

¥ ® X ¥ ®.

We denote X ¥ by hlimi ® ¥

X i¥, and call it the homotopy colimit of the sequence [2].

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 )):

X ¥ ® PiXi¥ ®1Ðshift PiXi

¥ ®.

We denote X ¥ by hlim¥ ¬ i

X i¥, and call it the homotopy limit of the sequence.

According to [2], for a complex X ¥ Î K (ModA ), we have the following isomorphisms in

D (ModA ):

hlimn ® ¥t³ -nX

¥ @ X ¥, hlim¥ ¬ n

s> -nX ¥

@ X ¥ , hlimn ® ¥s² nX

¥ @ X ¥ and hlim¥ ¬ n

t² nX ¥

@ X ¥ .

Spaltenstein, B�skstedt 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)

(X ¥, I ¥) @ HomK (ModA)(X ¥ , I ¥) for every complex I ¥ Î K s(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

HomD (ModA)(P ¥, X ¥) @ HomK (ModA)(P

¥ , X ¥) for every complex P ¥ Î K s(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).

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

complex provided that all X i 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

(C1) BUA¥ is contained in DmodA

b (ModA) as a right A- module complex , and is contained in

DBÐmodb (B- Mod) as a left B-module complex.

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

(C2l ) BUA¥ belongs to Kb(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;

(C4l ) the natural right multiplication morphism A op ® 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)):

(D1) F and G induce a duality between DmodAb (ModA ) and DBÐmod

b (B- Mod) ;

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

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

pair as functors between DmodAb (ModA ) and DBÐmod

b (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)

5

derived dual of B. Then there exists a ¶-functor G : D (B- Mod) ® D (ModA) which

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- Mod). By the above remark, for every complex Y ¥ in DBÐmodb (B- Mod), we have the

following isomorphisms.

Hi(GY ¥ ) @ HomD (ModA)(A,GY ¥ [i]) @ HomD(B-Mod)(Y ¥ ,FA[i])

@ HomD(ModA)(A, GÕY ¥[i]) @ Hi (GÕY ¥ ) for all i.

Then GY ¥ belongs to DmodAb (ModA ). Hence, by adjointness of F, G is isomorphic to GÕ as

functors from DBÐmodb (B- Mod) to DmodA

b (ModA ). Let l I(B) be the set of left ideals of B. In

case of B being left Noetherian, there exists some integer n such that we have the following

isomorphisms:

HomD(B-Mod)( ÅJ ÎlI(B )

B/J, FA[i]) @ PJ ÎlI(B )

HomD(B-Mod)(B/J , FA[i])

@ PJ ÎlI(B )

HomD(ModA)(A , GÕB/J[i])

@ HomD(ModA)(A , PJ ÎlI(B )

GÕB/J[i])

@ HomD(ModA)(A , GÕ( ÅJ ÎlI(B )

B/J)[i])

= 0 for all i > n.

By Lemma 3.1 (a), we get FA Î Db(B-Mod)fid . For every complex Y ¥ in D b(B- Mod), there

exist integers m ² n such that we have the following isomorphisms:

Hi(GY ¥) @ HomD(ModA )(A,GY ¥[i])

@ HomD(B-Mod)(Y ¥,FA[i])

= 0 for all i < m or i > n.

6

Then GY ¥ belongs to Db(ModA), and therefore the image of G|Db(B-Mod) is contained in Db(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

K s(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:

HomD(ModA)(X ¥, RHom¥

B(Y ¥, BUA

¥)) @ H0Hom¥A(P

¥, Hom¥B(Q ¥ , BUA

¥))

@ H0Hom¥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 X0¥ and Ui

¥ belong to U[r ] for all i ³ 1, such that X ¥ is isomorphic to hlim

¥ ¬ iX

i¥ in

7

D (ModA ). In case of U = addUA¥ for some complex UA

¥ of D [m,n](ModA), we simply call a

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.

(a) We have X k¥ Î D [s,t +k](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 .

(b) If A is a right coherent ring, and if U is a family of objects of D modA[m,n] (ModA), then

X ¥ belongs to D modA+ (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 {Xi ¥}i ³0 , then the induced natural morphism hlim

i ® ¥Hom¥

A(Xi ¥ , BUA

¥) ®

Hom¥A( hlim

¥ ¬ iXi

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

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

¥)

¯ ¯

hlimi ® ¥

Hom¥A(Xi

¥, BUA¥) h¾ ®¾ Hom¥

A( hlim¥ ¬ i

Xi ¥, BUA

¥).

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

isomorphisms:

HkHom¥A( hlim

¥ ¬ iXi

¥, BUA¥) @ HomD(ModA)( hlim

¥ ¬ iXi

¥, BUA¥[k ])

@ HomD(ModA)(s²t -k hlim¥ ¬ i

Xi ¥, BUA

¥[k ])

@ HomD(ModA)(s²t -kXp ¥, BUA

¥[k ]) for some p >> 0

8

@ HomD(ModA)(Xp ¥, BUA

¥[k ])

@ HkHom¥A(Xp

¥, BUA¥).

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

0:

HkHom¥A(Xq

¥ , BUA¥) @ HomD(ModA )(Xq

¥, BUA¥[k ])

@ HomD(ModA)(s²t -kXq ¥, BUA

¥[k ])

@ HomD(ModA)(s²t -kXq+j¥, BUA

¥[k ])

@ HkHom¥A(Xq+j

¥, BUA¥).

Then we have the isomorphism HkHom¥A(Xq

¥, BUA¥) @ Hk hlim

i ® ¥Hom¥

A(Xi ¥, BUA

¥). For all integers

r ³ max(p, q ), we have the following commutative diagram:

HkHom¥A(Xr

¥, BUA¥) = HkHom¥

A(Xr ¥, BUA

¥)

¯ ¯

Hk hlimi ® ¥

Hom¥A(Xi

¥, BUA¥) H kh¾ ®¾ HkHom¥

A( hlim¥ ¬ i

Xi ¥ , BUA

¥),

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

complex satisfying the conditions (C1), (C2r ), (C3r ) and (C4r ). Then { D B modÐ (B- Mod);

Hom¥A(-, BUA

¥), R -Hom¥B(-, BUA

¥)} is a localization duality of D modA+ (ModA ), and the image

of HomA(-, BUA¥)|Db

modA(ModA ) is contained in D B modb (B- Mod). Moreover, every complex in

D modA+ (ModA ) is a UA

¥-limit complex if and only if Hom¥A(-, BUA

¥) and R -Hom¥B(-, BUA

¥)

induce the duality between D B modÐ (B- Mod) and D modA

+ (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

¥) :

9

D modA+ (ModA) ® D B mod

Ð (B- Mod), and the image of R +HomA(-, BUA¥)|Db

modA(ModA ) is contained

in D B modb (B- Mod). It is clear that R -Hom¥

B(-, BUA¥) : D B mod

Ð (B- Mod) ® D +(ModA )

exists. Since D B modÐ (B- Mod) is equivalent to D -(B- mod), D B mod

Ð (B- Mod) is equivalent

to K -(BP). Given a complex X ¥ Î D B modÐ (B- Mod), there exists a complex P ¥ Î K -(BP)

such that X ¥ is isomorphic to P ¥ in D B modÐ (B- Mod). Since R -Hom¥

B(P ¥ , BUA

¥) is isomorphic

to R -Hom¥B( hlimn ® ¥t³-nP

¥, BUA¥) @ hlim¥ ¬ n

R -Hom¥B(t³-nP

¥, BUA¥), R -Hom¥

B(P ¥, BUA

¥) is a UA¥-limit

complex. By Lemma 2.4, R -Hom¥B(P

¥, BUA¥) is contained in D modA

+ (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):

hlimn ® ¥t³-nP

¥ ® hlimn ® ¥R +Hom¥

A(R -Hom¥B(t³-nP

¥, BUA¥), BUA

¥)

¯

R +Hom¥A( hlim¥ ¬ n

R -Hom¥B(t³-nP

¥ , BUA¥), BUA

¥)

¯

hlimn ® ¥t³-nP

¥ ® R +Hom¥A(R -Hom¥

B( hlimn ® ¥t³-nP

¥, BUA¥), BUA

¥)

¯ ¯

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

D B modÐ (B- Mod) and D modA

+ (ModA ), then every complex in D modA+ (ModA ) is a UA

¥-limit

complex. Conversely, if every complex X ¥ in D modA+ (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¥ ® ,

10

¼ ,

where X0¥ and Ui

¥ belong to (addUA¥ )[r ] for all i ³ 1, such that X

¥ is isomorphic to hlim¥ ¬ i

X 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¥ ¬ i

Xi¥ ® Hom¥

B(Hom¥A( hlim

¥ ¬ i

Xi¥, BUA

¥), BUA¥) is an isomorphism in D (ModA). Therefore, Hom¥

B(-, BUA¥): D B mod

Ð (B- Mod)

® D modA+ (ModA ) is dense, and hence Hom¥

A(-, BUA¥) and Hom¥

B(-, BUA¥) induce the duality

between D B modÐ (B- Mod) and D modA

+ (ModA ).

Corollary 2.7. Let A be a right coherent ring, B a left coherent ring, BUA¥ a B-A-bimodule

complex satisfying the conditions (C1), (C2r ), (C2l ), (C3r ) and (C4r ). Then { D B modb (B-

Mod); Hom¥A(-, BUA

¥), Hom¥B(-, BUA

¥)} is a localization duality of D modAb (ModA ).

Proof. By the condition (C2l ), it easy to see that the image of R bHom¥B(-, BUA

¥) @

Hom¥B(-, BUA

¥) is contained in D modAb (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

11

B-A-bimodule complex which satisfies the conditions (C1) and (C2r ). Assume that the

image of Hom¥A(-, BUA

¥) : D modAÐ (ModA ) ® D BÐmod

+ (B- Mod) contains B- mod. If E ¥ is a

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

Åi ³ s

E i is a finitely embedding cogenerator for B- mod, and then Pi ³ s

E i is a finitely embedding

injective cogenerator for B- mod.

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

Ð (ModA ) is equivalent

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

K -(sumAA) such that Hom¥A(P ¥, BUA

¥) is isomorphic to X in D b(B- Mod). Since Hom¥A(P ¥,

BUA¥) is BU

¥-limit complex, there exists an integer n such that we have isomorphism HiHom¥A(P

¥,

BUA¥) @ HiHom¥

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

P m , where P i Î sumAA (n ² i ² m ). Then HomA(P i, 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( Åi ³ s

E i). Then we have the

following exact sequences:

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

0 ® Imd-1 ® Kerd0 ® X ® 0 ...........................................(2).

Since I 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( Åi ³ s

E i ), X is embedded in some finite

direct sum of copies of Åi ³ s

E n.

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

complex BUA0® ¼ ® BUA

n satisfying the conditions (C1), (C2r ), (C2l ), (C3r ) and (C4r ).

Then Å

i 0

nU i is a finitely embedding injective cogenerator for B- mod.

Proof. By Corollary 2.7 and Theorem 2.9.

12

Corollary 2.11. Let A be a right coherent ring, B a left coherent ring, BUA¥ a cotilting

B-A-bimodule complex BUA0® ¼ ® BUA

n . Then Å

i 0

nU i is a finitely embedding injective

cogenerator for B- mod.

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

B-A-bimodule complex BUA0® ¼ ® BUA

n . Then every injective indecomposable left

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

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

¥Ä·

BHom¥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

(T1) ATB¥ belongs to Kb(PB) as a right B- module complex, and belongs to Kb(AP) as a left

A-module complex.

In case of finite dimensional k- algebras over a field 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

Proof. It is clear that Hom¥R(ATB

¥ , w ¥ ) is contained in Kb(InjA) as a right A- module

complex, and is contained in Kb(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

D modRb (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-

module and as a left B- module. Therefore, Hom¥R(ATB

¥ , w ¥ ) is contained in DmodA

b (ModA)

as a right A- module complex, and is contained in D B modb (B- Mod) as a left B-module

complex. In order that Hom¥R(ATB

¥ , w ¥ ) satisfies the conditions (C3r ) and (C4r ), it suffices

to show that the natural morphism AA ® R bHom¥B(R bHom¥

A(AA, Hom¥R(ATB

¥ , w ¥ )), Hom¥

R(ATB¥,

w ¥ )) is an isomorphism in DmodAb (ModA). By Lemma 2.13, we have the following isomorphisms

in D (ModA ):

R bHom¥B(R bHom¥

A(AA , Hom¥R(ATB

¥ , w ¥ )), Hom¥

R(ATB¥ , w ¥ ))

@ Hom¥B(Hom¥

R(TB¥ , w ¥ ), Hom¥

R(ATB¥ , w ¥ ))

@ Hom¥R(ATB

¥Ä·

BHom¥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

14

finitely generated as an R-module. If w ¥ is a dualizing R-module complex, then

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

projective (resp., injective) B opÄkA- 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:

R Hom¥A(-,-) : D (ModB opÄkA )op´D (ModC opÄkA )®D (ModC opÄkB ),

-

Ä·AL

- : D (ModB opÄkA )´D (ModAopÄkC )®D (ModB opÄkC ).

Let Db(ModA )fid be the triangulated subcategory of Db(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 HomDb(ModA )( ÅI ÎrI(A )

A/I, X ¥ [i ]) = 0 for all i > n ,

then X ¥ belongs to Db(ModA )fid .

(b) In case of A being a right Artinian ring, if there exist an integer n such that

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

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

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

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

15

complex, and belongs to K b(B- Inj) as a left B-module complex, such that BUA¥ is isomorphic

to BVA¥ in D (ModB opÄkA ).

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:

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

(C2 l ) BUA¥ belongs to D b(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

complex GB Î D modAb (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.

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

16

B-A- bimodules:

Hi(BUA¥) @ HomD (ModA ) (A, BUA

¥[i ])

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

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

Since FA belongs to D B modb (B- Mod), then BUA

¥ belongs to D B modb (B- Mod). Therefore

BUA¥ satisfies the condition (C1). Since Å

I ÎrI(A )A/I belongs to D b(ModA ), F( Å

I ÎrI(A )A/I )

belongs to D b(B- Mod). Then there exists an integer n such that we have

HomD (ModA )( ÅI ÎrI(A )

A/I , BUA¥[i ]) @ HomD (ModA )( Å

I ÎrI(A )A/I , GB [i ])

@ HomD (B-Mod)(B, F( ÅI ÎrI(A )

A/I ) [i ])

= 0 for all i > n.

By Lemma 3.1 (a), we get BUA¥ Î D modA

b (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¥B( ÅJ ÎlI(B )

B/J , BUA¥) @ P

J ÎlI(B )R Hom¥

B(B/J , BUA¥)

@ PJ ÎlI(B )

G(B/J )

@ G( ÅJ ÎlI(B )

B/J ).

Then R Hom¥B( ÅJ ÎlI(B )

B/J , BUA¥) belongs to D b(ModA ). Since {R Hom¥

A(-, BUA¥) :

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

HomD (B- Mod)( ÅJ ÎlI(B )

B/J , BUA¥[i ]) @ HomD (B- Mod)( Å

J ÎlI(B )B/J , R Hom¥

A(AA, BUA¥)[i ])

@ HomD (ModA)(AA, R Hom¥B( ÅJ ÎlI(B )

B/J , BUA¥)[i ])

= 0 for all i > n .

By Lemma 3.1 (a), we get BUA¥ Î D B mod

b (B- Mod)fid . Since the natural morphism B ®

17

R Hom¥A(BUA

¥, BUA¥) is an isomorphism in D b(ModB opÄkB ), we have an isomorphism P ®

R Hom¥B(R HomA(P, BUA

¥), BUA¥), for every finitely generated projective left B- module. Then

we have an isomorphism P ¥ ® R Hom¥A(R Hom¥

B(P ¥, BUA

¥), BUA¥) for every P ¥ Î K b(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(t ³-nQ

¥, BUA¥)}, we have

the following isomorphisms in D (B- Mod):

hlimn ® ¥t³-nQ

¥ ® hlimn ® ¥R Hom¥

A(R Hom¥B(t³-nQ

¥, BUA¥), BUA

¥)

¯

R Hom¥A( hlim¥ ¬ n

R Hom¥B(t³-nQ

¥, BUA¥), BUA

¥)

¯

hlimn ® ¥t³-nQ

¥ ® R Hom¥A(R Hom¥

B( hlimn ® ¥t³-nQ

¥, BUA¥), BUA

¥)

¯ ¯

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 ).

Lemma 3.4. Let UA¥ be a complex in D b(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 ).

Proof. Let H := R Hom¥B(-, BUA

¥), P ¥ a complex ¼® P -1® P 0 ® 0 ®¼ which belongs

18

to K -(BP), and Pi¥ := t³- iP

¥. Then we have the following sequences of distinguished triangles:

GP -1[-1] ® GP1¥ ® GP

0 ®, HP -1[-1] ® HP1¥ ® HP

0 ®,

GP -2[-2] ® GP2¥ ® GP1

¥ ®, HP -2[-2] ® HP2¥ ® HP1

¥ ®,

¼ ¼

GP -i[-i ] ® GPi¥ ® GPi-1

¥® , HP -i[-i ] ® HPi¥ ® HPi-1

¥® ,

¼ , ¼ .

By inductive step, we construct isomorphisms between distinguished triangles:

GP -i[-i ] zi¾ ®¾ GPi¥ yi¾ ®¾ GPi-1

¥ xi¾ ®¾ GP -i[-i +1]

¯bi ¯ai ¯ai-1 ¯bi[1]

HP -i[-i ] wi¾ ®¾ HPi¥ vi¾ ®¾ HPi-1

¥ ui¾ ®¾ HP -i[-i +1],

where P0¥ := P

0. 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

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 - i 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

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

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

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

Pi¥ @ hlim

¥ ¬ iGPi

¥ @ hlim¥ ¬ i

HPi¥ @ H hlim

i ® ¥Pi

¥ @ HP ¥ .

Remarks. The conditions (D2r ) and (D2l ) are closely related to the property of finite

19

injective dimension of complexes. Indeed, let R be a commutative Noetherian regular ring

of infinite Krull dimension, and A := R [X]/(X2 - a ), where a is a non-zero element in N 2 for

some maximal ideal 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,

but is not a dualizing complex. Moreover, A induces a duality D modAb (ModA ) ® D modA

b

(ModA ) (oral communication with Y. Yoshino). I donÕt know if an arbitrary locally Gorenstein

ring A induces a self-duality on D modAb (ModA ), or equivalently if for every prime ideal P of

an arbitrary locally Gorenstein ring A, there is some integer n such that ExtAi (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

3.3, it suffices to show that BUA¥ belongs to D modA

b (ModA )fid and D B modb (B- Mod)fid .

Since A is right Artinian and B is left Artinian, in the proof of Theorem 3.3, we can replace

ÅI ÎrI(A )

A/I and ÅJ ÎlI(B )

B/J by A /radA and B /radB , respectively. We are done by Lemma

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

complex in D +(ModB opÄkA ). 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

BUA¥ÄAL [n ] in D +(ModB opÄkA ).

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

¥) :

D modAb (ModA ) ® D modA

b (ModA ) are derived equivalences. Since D modAb (ModA ) @

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

Hom¥A(Hom¥

B(-, BUA¥), BVA

¥) @ -

Ä·AL

Hom¥B(BUA

¥, BVA¥),

Hom¥A(Hom¥

B(-, BVA¥), BUA

¥) @ -

Ä·AL

Hom¥B(BVA

¥, BUA¥).

Let M ¥ and N ¥ be A- bimodule complexes Hom¥B(BUA

¥, BVA¥) and Hom¥

B(BVA¥, BUA

¥), respectively.

It is clear that M ¥ and N ¥ are contained in D modAb (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 ])

@ HiR Hom¥A(NA

¥, AAA ) for all i .

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

b (A- Mod). Also,

M ¥

Ä·AL

N ¥ and N ¥

Ä·AL

M ¥ are isomorphic to A in D (ModA opÄkA ). 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

have Hp(M ¥)ÄAHq(N ¥) @ Hp+q(M ¥

Ä·AL

N ¥) and Hq(N ¥)ÄAHp(M ¥) @ Hp+q(N ¥

Ä·AL

M ¥). Let X :=

Hp(M ¥) and Y := Hq(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

are nonzero. By locality of A , X /X (radA ) ÄAY /(radA )Y is non-zero, and Hp(M ¥)ÄAHq(N ¥)

is non-zero. Similarly, Hq(N ¥)ÄAHp(M ¥) is non-zero. Then p +q = 0 and Hp(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

A @ M ¥

Ä·AL

N ¥

@ L [q ]

Ä·AL

L* [-q ]ÅL [q ]

Ä·AL

NÕ ¥ÅMÕ ¥

Ä·AL

L* [-q ]ÅMÕ ¥

Ä·AL

NÕ ¥.

Then L [q ]

Ä·AL

NÕ ¥ÅMÕ ¥

Ä·AL

L* [-q ]ÅMÕ ¥

Ä·AL

NÕ ¥ is acyclic, and MÕ ¥ and NÕ ¥ are acyclic.

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

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

BVA¥ @ Hom¥

A(Hom¥B(BUA

¥, BUA¥), BVA

¥)

@ BUA¥

Ä

·AL

Hom¥B(BUA

¥, BVA¥)

@ BUA¥

Ä

·AL

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:

1 2·¾ ®¾¬ ¾¾ ·

a

b,

with aba = bab = 0. Then A , A e1Äke1A ® A and A e2Äke2A ® A are dualizing A-

bimodule complexes, where morphisms are natural multiplications.

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22

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