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TILTING MODULES OF FINITE PROJECTIVE DIMENSION: SEQUENTIALLY STATIC AND COSTATIC MODULES

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DIMENSION:

SEQUENTIALLY STATIC AND COSTATIC MODULES

ALBERTO TONOLO

Abstract. In [5], Miyashita introduced tilting modules of finite pro- jective dimension. A tilting moduleAV of projective dimension less or equal than r furnishes r+1 equivalences between subcategories of A-Mod and End V -Mod: we call static and costatic the modules in A-Mod and End V -Mod, respectively, involved in these equivalences. In this paper we characterize the modules in A-Mod and End V -Mod which have a filtration with static and costatic factors, respectively.

Introduction

Tilting modules arose from representation theory of algebras and are known to furnish equivalences between categories of modules. Their def- inition has its origin in the works of Gel’fand Ponomarev, Brenner and Butler, Happel and Ringel (see [1] for a good reference); since then there have been generalizations in several directions. One of these is the study of tilting modules for arbitrary rings.

Let A be an arbitrary associative ring. A left A-module V is said tilting provided it satisfies the following properties:

(1) There is an exact sequence 0 → AP1AP0AV → 0, with P1

and P0 finitely generated projective modules (thus, pdAV ≤ 1).

(2) Ext1A(V, V ) = 0.

(3) There is an exact sequence 0 →AA →AV1AV0 → 0, with V1 and V0 summands of a finite direct sum of copies of V .

The main result on tilting modules is essentially due to Brenner and Butler [2]. Stated for a tilting module over a finite dimensional algebra, it has been later generalized by Colby and Fuller [3] to tilting modules over arbitrary associative rings.

Theorem of Brenner-Butler Let AV be a tilting module, and B = EndAV . Then VBalso is a tilting module, and A ∼= End VB. Moreover there are two category equivalences

Gen(AV ) = Ker(Ext1A(V, −))

HomA(V,−)

−−−−→

←−−−−

VB⊗− Ker(TorB1(V, −)) and Ker(HomA(V, −))

Ext1A(V,−)

−−−−→

←−−−−

TorB1(V,−)

Ker(VB⊗ −).

1

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The pairs

(Ker(Ext1A(V, −)), Ker(HomA(V, −))) and (Ker(VB⊗−), Ker(TorB1(V, −))) are torsion theories in A-Mod and B-Mod, respectively.

Denote by H, H(1), T and T(1) the functors HomA(V, −), Ext1A(V, −), VB⊗− and TorB1(V, −). We call static the 0-static and the 1-static modules, i.e. the left A-modules in Gen(AV ) = Ker(Ext1A(V, −)) and Ker(HomA(V, −)), respectively. Analogously, we call costatic the 0-costatic and the 1-costatic modules, i.e. the left B-modules in Ker(TorB1(V, −)) and Ker(VB⊗ −), re- spectively.

For each left A-module M (resp. each left B-module N ) we have the following short exact sequence (see [3, Theorem 1.4])

0 → T H(M ) → M → T(1)H(1)(M ) → 0 (resp. 0 → H(1)T(1)(N ) → N → HT (N ) → 0),

where T H(M ) is 0-static and T(1)H(1)(M ) is 1-static (resp. HT (N ) is 0- costatic and H(1)T(1)(N ) is 1-costatic). In other words we have the filtrations

M ≥ T HM ≥ 0 (resp. N ≥ H(1)T(1)N ≥ 0)

with 1- and 0- static (resp. 0- and 1- costatic) filtration factors. Thus all modules in A-Mod and B-Mod consist sequentially of pieces of static and costatic modules, respectively; in particular each simple left A-module is static, each simple left B-module is costatic.

In [5] Yoichi Miyashita introduced tilting modules of finite projective di- mension over an arbitrary associative ring A. He proved various fundamental results, in particular a generalization of the Brenner and Butler Theorem.

Following his definition, a left A-moduleAV is said to be a tilting module of projective dimension ≤ r if it satisfies the following three conditions:

(1) AV has a projective resolution

0 →APr→ . . . →AP0AV → 0 with each Pi finitely generated.

(2) ExtiA(V, V ) = 0, if 1 ≤ i ≤ r.

(3) There exists an exact sequence

0 →AA →AV0AV1 → . . . →AVr→ 0

with eachAVi summand of a finite direct sum of copies ofAV . If r = 0 then AV is a progenerator, if r = 1 then AV is a tilting module.

Theorem of Miyashita LetAV be a tilting module of projective dimen- sion ≤ r, and B = EndAV . Then VB also is a tilting module of projective dimension ≤ r, and A ∼= End VB. There are r + 1 category equivalences

KEe(AV )

ExteA(V,−)

−−−−→

←−−−−

TorBe(V,−)

KTe(VB), 0 ≤ e ≤ r

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where

KEe(AV ) = {M ∈ A-Mod : ExtiA(V, M ) = 0, if 0 ≤ i ≤ r and i 6= e}

KTe(VB) = {M ∈ B-Mod : TorBi (V, M ) = 0, if 0 ≤ i ≤ r and i 6= e}.

We say e-static the modules in KEe(AV ) and e-costatic the modules in KTe(VB); the left A-modules in the classes KEe(AV ), e ≥ 0, are called static, the left B-modules in the classes KTe(VB), e ≥ 0, are called costatic.

Increasing from 2 to r + 1 the number of category equivalences, we loose the possibility to interpretate them in terms of torsion theories. Not only, but modules are not more made sequentially of pieces of static or costatic modules: in particular, we have examples of simple modules which are nei- ther static nor costatic (see Example 1.5).

Roughly speaking, the main result of this paper characterize the left A- modules (resp. the left B-modules) which are sequentially made of pieces r-, r − 1-,. . . , 1-, 0- static modules (resp. 0-, 1-, . . . , r − 1-, r- costatic modules).

Let us be more precise. We say that a left A-module M (a left B-module N ) is sequentially static (resp. sequentially costatic) if for each i 6= j ≥ 0,

T(i)H(j)M = 0 (resp. H(i)T(j)M = 0).

This definition is justified by our main result:

Theorem (1.10,1.11) LetAV be a tilting module of projective dimension

≤ r, and B = EndAV . Then a left A-module M (resp. a left B-module N ) is sequentially static (resp. costatic) if and only if there exists a filtration

M = Mr ≥ Mr−1≥ Mr−2≥ . . . ≥ M0 ≥ M−1 = 0 (resp. there exists a filtration

N = N−1≥ N0≥ N1≥ . . . ≥ Nr−1≥ Nr= 0)

such that Me/Me−1 is e-static (resp. Ne−1/Ne is e-costatic) for each e ≥ 0.

In such a case Me/Me−1 ∼= T(e)H(e)M (resp. Ne−1/Ne ∼= H(e)T(e)N ) for each e ≥ 0.

We are pleased to finish the introduction, underlining the striking analogy between our main theorem and the characterization of sequentially Cohen- Macaulay modules over a commutative Gorenstein ring due to Christian Peskine (see [6, Ch. III, Theorem 2.11], [4, Theorem 1.4]): In fact it is its covariant version, in the more general setting of modules over an arbitrary associative ring, with the tilting module playing the role of the Gorenstein ring and the Cohen-Macaulay condition substituted by its homological char- acterization.

1. The Main Theorem

Let A be an associative ring and AV be a left tilting A-module of pro- jective dimension ≤ r (see Introduction). Denote by B the endomorphism ring EndAV . By [5, Theorem 1.5], AVB is a faithfully balanced bimodule and VB is a right tilting module of projective dimension ≤ r.

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Denote by H = H(0), H(i), T = T(0) and T(i) the functors HomA(V, −), ExtiA(V, −), VB⊗ − and TorBi (V, −), respectively.

Lemma 1.1. [5, Lemma 1.7]

(i) For any injective left A-module I, there is a canonical isomorphism T HI → I, t ⊗ f 7→ f (t).

(ii) For any projective left B-module P , there is a canonical isomorphism P → HT P, p 7→ (t 7→ t ⊗ p).

(iii) If i ≥ 1, then for all injective left A-module I and all projective left B-module P we have

T(i)HI = 0 and H(i)T P = 0.

We concentrate our attention on left A-modules. Analogous results can be obtained, with the obvious modifications, on the category of left B-modules.

Let us fix a left A-module M and an injective resolution 0 → M → I with differential operator d. We denote by Ji, i ≥ 0, the ith-syzygy of I, i.e. the kernel of di+1: Ii+1→ Ii+2; J−1 will indicate the module M itself.

As usual, we denote by Bi(M ) and Zi(M ), i ≥ 0, the i-boundaries and the i-cycles of the complex

0 → H(I) := 0 → HI0 → HI1 → HI2→ . . . . We have the following exact sequences

(1) 0 → HM → HI0 → B1(M ) → 0.

(2) 0 → Bi(M ) → Zi(M ) → H(i)M → 0, i ≥ 1.

(3) 0 → Zi(M ) → HIi → Bi+1(M ) → 0, i ≥ 1.

Applying the functor T , by Lemma 1.1 we get T1. . . . → 0 → T(i+1)B1(M ) → T(i)HM → 0 →

. . . → 0 → T(2)B1(M ) → T(1)HM → 0 → (0 →)T(1)B1(M ) → T HM → T HI0 ∼= I0→ T B1(M ) → 0.

T2. . . . → T(j)Bi(M ) → T(j)Zi(M ) → T(j)H(i)M → . . .

. . . → T(1)H(i)M → T Bi(M ) → T Zi(M ) → T H(i)M → 0.

T3. . . . → 0 → T(j+1)Bi+1(M ) → T(j)Zi(M ) → 0 → . . .

→ 0 → T(1)Bi+1(M ) → T Zi(M ) → T HIi∼= Ii→ T Bi+1(M ) → 0.

From these exact sequences we obtain easily the solid part of the following commutative diagram with exact rows and columns. The dotted arrows are usual factorizations of morphisms, or compositions, or easy consequences.

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(Diagram (A))

0

T(1)B1(M )

 0

T H(M )



wwwwo o

T B2(M )

OO

C1 u P ''P P

0_ _ _ _ //_M _ _ _ _ _ //_T HI0 ∼= I0 //

ξ

O ''O O O O O O



T HI1∼= I1

OO

C2 t

&&N NN T(1)Z1(M ) //T(1)H(1)M //

88 88p p

T B1(M ) //



T Z1(M )

OO //T H(1)M //0

0

T(2)Z2(M ) //T(2)H(2)M //T(1)B2(M )

OO //T(1)Z2(M ) //T(1)H(2)M

0

OO

. . . .

T(i)Zi(M ) //T(i)H(i)M //T(i−1)Bi(M ) //T(i−1)Zi(M ) //T(i−1)H(i)M Lemma 1.2. Given the solid part of the commutative diagram

0

 0



C

α C



0 //A //

β M ϕ //

ϑ

L

0 //B //N ψ //L

with exact rows and columns, there are unique maps α and β such that the diagram commutes. With these maps the first column is exact; moreover, if ϑ is epic, then so is β.

Proof. This follows by diagram chasing.

Applying Lemma 1.2 to the commutative triangles from Diagram A T HI0 ∼= I0



ξ

N ''N NN NN NN NN N

T B1(M ) //T Z1(M )

and T HI0 ∼= I0 //

ξ

P ''P PP PP PP PP

PP T H(I1) ∼= I1

T Z1(M )

OO ,

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we get the following exact sequences

0 → C1 → Ker ξ → C2 → 0 and 0 → Ker ξ → M → Tη (1)B2(M );

if T H(1)M = 0, the morphism η is epic. Since T(i)Zi+1(M ) ∼= T(i+1)Bi+2(M ), i ≥ 1 (see T 3), componing with morphisms from Diagram (A), we obtain the following diagram with exact rows and columns and natural morphisms.

It describes how the functors 1Mod, T(i)H(i), i ≥ 0, are canonically related.

(Diagram (B))

T(1)H(2)M

0 . . . //T(1)Z2(M ) ∼= T(2)B3(M ) //

OO

. . .

0 C2

OO

oo T(1)H(1)Moo T(1)Z1(M )oo

0 // Ker ξ

OO //M η //T(1)B2(M )

OO

T HM

OO ρM 66nnnnnnnnnn

T(2)H(2)M

OO

T(1)B1(M )

OO

T(2)Z2(M )

OO

0

OO

The map ρM is the counity of the adjunction between the functors T and H.

Definition 1.3. We say that a left A-module (resp. a left B-module) is e-static (resp. e-costatic) if it belongs to the class

KEe(AV ) = {M ∈ A-Mod : ExtiA(V, M ) = 0, i 6= e}

(resp. KTe(VB) = {N ∈ B-Mod : TorBi (V, N ) = 0, i 6= e}).

The e-static and the e-costatic modules, for some e ≥ 0, are called static and costatic modules, respectively.

In [5, Theorem 1.16], Miyashita proved that the categories of e-static and that of e-costatic modules “are equivalent under the functors ExteA(V, −) and TorBe(V, −), which are mutually inverse to each other”.

Definition 1.4. We say that a left A-module M (a left B-module N ) is sequentially static (costatic) if for each i 6= j ≥ 0,

T(i)H(j)M = 0 (H(i)T(j)N = 0).

Why these modules are called sequentially static and costatic will be clear after our Theorems 1.10, 1.11.

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If the projective dimension of the tilting moduleAT is less or equal than 1, then any left A-module is sequentially static and any left B-module is sequentially costatic ([3, Theorem 1.4]). This is not more true for tilting modules of higher projective dimension:

Example 1.5. In this example, k denotes an algebraically closed field, and all rings are finite-dimensional k-algebras given by quivers. If i is a vertex of a quiver, we denote by P (i) the indecomposable projective associated to i, by E(i) the indecomposable injective associated to i, and we denote by S(i) the simple top of P (i), or, equivalently, the simple socle of E(i).

(1) Let A denote the k-algebra given by the quiver 1 → 2a → 3 withb relation ba = 0. It is easy to verify that the minimal injective cogenerator

AV = 2 3 ⊕ 1

2 ⊕ 1 is a tilting left A-module of projective dimension 2.

The ring B = EndAV is the k-algebra given by the quiver 4→ 5c → 6 withd relation dc = 0. With some calculation, it is possible to see that

T(2)H(1)S(2) ∼= T(2)S(4) ∼= T(1)S(5) ∼= S(3) 6= 0.

Observe, moreover, that HS(2) ∼= S(6) 6= 0 and hence the simple module S(2) is not static, since it does not belong to any class KEe(AV ), e = 0, 1, 2.

(2) Let R denote the k-algebra given by the quiver

a 1

 ==b=

2

c 3

4

with the relation ca = 0. It is easy to verify that the minimal injective cogeneratorRV = 2

4 ⊕ 1 2 ⊕ 1

3 ⊕ 1 is a tilting left R-module of projective dimension 2. The ring S = EndRV is the k-algebra given by the quiver

d 5

 ==e=

7

f 6

8

with the relation fd = 0. With some calculation, it is possible to see that HT(1)S(5) ∼= HE(3) ∼= S(6) 6= 0.

Observe, moreover, that T(2)S(5) ∼= T(1)S(7) = S(4) 6= 0; hence the simple left B-module S(5) is not costatic, since it does not belong to any class KTe(VB), e = 0, 1, 2.

Lemma 1.6. Let Q be an injective cogenerator of the category of left A- modules. We denote by V the left B-module HQ. For each left A-module M and each left B-module N we have:

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(1) Im H ⊆ Cogen V ⊆ Ker T(r) and Im T ⊆ Gen V ⊆ Ker H(r). (2) If r ≥ 2, T(r−1)HM = 0 and H(r−1)T N = 0.

(3) T H(r)M = 0 and HT(r)N = 0.

Proof. 1. Let us consider the exact sequences

0 →AM → QX and B(X)BN → 0.

Applying the functors H and T we get

0 → HM → V∗X and V(X) → T N → 0.

Therefore HM and T N are respectively cogenerated by Vand generated by V . Let us assume M0 and N0 respectively cogenerated by V and generated by V . Now T(r+1) and H(r+1) are the zero functors; then applying T(r) and H(r) we get

0 → T(r)M0 → T(r)V∗X and H(r)V(X)→ H(r)N0→ 0.

We conclude since by Lemma 1.1, (iii),

T(r)V∗X= TrH(QX) = 0 and H(r)V(X)= HrT (B(X)) = 0.

2. Let 0 → M → I a P → N → 0 be an injective and a projective resolutions. Applying Tr−1 and Hr−1 to the exact sequences

0 → HM → HI0→ B1(M ) → 0 and 0 → B0(N ) → T P0 → T N → 0 we have

T(r)B1(M ) → Tr−1HM → Tr−1HI0= 0 and 0 = H(r−1)T P0 → H(r−1)T N → H(r)B0(N ).

Since B1(M ) is cogenerated by Vand B0(N ) is generated by V we conclude.

3. Let us prove T H(r)M = 0. Consider an injective resolution 0 → M → I; denote by Ji the ith syzygy. It is easy to verify that H(r)M ∼= H(1)Jr−2. By [5, Lemma 1.1], Jr−1 belongs to KE0(V ) and hence HJr−1 belongs to KT0(V ) (see [5, Lemma 1.8]). From the commutative diagram with exact rows

T HIr−1 //

=



T HJr−1 //

=



T H(1)Jr−2 ∼= T H(r)M //

 0

Ir−1 //Jr−1 //0

we get T H(r)M = 0. In the same way we can prove HTrN = 0 for each left B-module N .

Lemma 1.7. If T(j+1)H(j)M = 0, for each j ≥ 1, then TiZi(M ) = 0 for each i ≥ 1. Moreover T1B1(M ) ∼= T1Z1(M ) = 0.

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Proof. Since Zr(M ) ≤ HIr ∈ Cogen V, by Lemma 1.6, (i), we have T(r)Zr(M ) = 0. Assume T(j)Zj(M ) = 0, j ≥ 2; from the exact sequence (T2)

0 = T(j+1)H(j)M → T(j)Bj(M ) → T(j)Zj(M ) = 0 we get T(j)Bj(M ) = 0 and hence, by T 3, T(j−1)Zj−1(M ) = 0.

Theorem 1.8. Let M be a left A-module such that T(j)H(j+1)M = 0 = T(i+1)H(i)M for each j ≥ 0 and i ≥ 0. Then there exists a filtration

M = Mr ≥ Mr−1≥ Mr−2≥ . . . ≥ M0 ≥ M−1 = 0 with the filtration factors Mi/Mi−1 isomorphic to T(i)H(i)M , i ≥ 0.

Proof. By hypothesis and Lemma 1.7, Diagram B becomes (Diagram (C))

0 0oo T(r)H(r)M T(r−2)Br−1(M )

OO

oo T(r−1)H(r−1)Moo 0oo

ηr−2

OO

0

0 . . .oo η3 T(2)B3(M )

OO

T(3)H(3)M

oo 0oo

T(1)H(1)M

OO

0 // Ker ξ

OO //M η //T(1)B2(M )

η2

OO //0

T HM

OO

T(2)H(2)M

OO

0

OO

0

OO

Let us consider now

Mr−1 := (ηr−2. . .η2η)−1(T(r−1)H(r−1)M ) Mr−2 := (ηr−3. . .η2η)−1(T(r−2)H(r−2)M )

. . .

M2 := η−1(T(2)H(2)M ) M1:= Ker ξ M0 := T HM.

It is easy to prove that in such a way we have constructed a filtration of M satisfying the thesis.

Analogously we get

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Theorem 1.9. Let N be a left B-module such that H(j+1)T(j)N = 0 = H(i)T(i+1)N for each j ≥ 0 and i ≥ 0. Then there exists a filtration

N = N−1 ≥ N0 ≥ N1 ≥ . . . ≥ Nr−1 ≥ Nr= 0 with the filtration factors Ni−1/Ni isomorphic to H(i)T(i)N , i ≥ 0.

The above results can be specialized in our main theorems.

Theorem 1.10. Let AV be a tilting module of projective dimension ≤ r, and B = EndAV . Then a left A-module M is sequentially static if and only if there exist a filtration

M = Mr ≥ Mr−1≥ Mr−2≥ . . . ≥ M0 ≥ M−1 = 0

with e-static filtration factors Me/Me−1, e ≥ 0. In such a case the filtration factor Me/Me−1 is isomorphic to T(e)H(e)M for each e ≥ 0.

Proof. Let M be a sequentially static module. Applying Theorem 1.8 we construct the filtration. Moreover, by hypothesis, Mi/Mi−1∼= T(i)H(i)M belongs to KEi(V ).

Conversely, assume there exists a filtration

M = Mr ≥ Mr−1≥ Mr−2≥ . . . ≥ M0 ≥ M−1 = 0

with Mi/Mi−1 in KEi(V ) for each i ≥ 0. Let us consider the short exact sequences

0 → Mi−1→ Mi → Mi/Mi−1→ 0, i ≥ 1.

Fix j ≥ 0. First, if j ≥ 1, H(j)Ml= 0 for each l < j: we have H(j)M0 = H(j)T HM = H(j)(M0/M1) = 0.

Assume by induction 0 = H(j)Ml−1; then we have

0 = H(j)Ml−1 → H(j)Ml→ H(j)(Ml/Ml−1) = 0, and hence H(j)Ml= 0. Second, applying H(j) we get

H(j−1)(Mi/Mi−1) → H(j)Mi−1→ H(j)Mi → H(j)(Mi/Mi−1) → H(j+1)Mi−1; if j = i we obtain

H(j)Mj ∼= H(j)(Mj/Mj−1),

if j < i we obtain H(j)Mi−1∼= H(j)Mi. Therefore, for each j ≥ 0, we have H(j)M = H(j)Mr∼= . . . ∼= H(j)Mj ∼= H(j)(Mj/Mj−1).

Since Mj/Mj−1belongs to KEj(V ), by [5, Theorem 1.14] the module H(j)M ∼= H(j)(Mj/Mj−1) belongs to KTj(V ) and hence T(i)H(j)M = 0 if i 6= j: then M is sequentially static.

Analogously we can prove

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Theorem 1.11. Let AV be a tilting module of projective dimension ≤ r, and B = EndAV . Then a left B-module N is sequentially costatic if and only if there exist a filtration

N = N−1 ≥ N0 ≥ N1 ≥ . . . ≥ Nr−1 ≥ Nr= 0

with e-costatic filtration factors Ne−1/Ne, e ≥ 0. In such a case the filtration factor Ne−1/Ne is isomorphic to H(e)T(e)N for each e ≥ 0.

There is a striking analogy between our Theorems 1.10, 1.11 and the characterization of sequentially Cohen-Macaulay modules over a commuta- tive Gorenstein ring due to Christian Peskine (see [6, Ch. III, Theorem 2.11], [4, Theorem 1.4]): in fact they are its covariant version in the more general setting of modules over an arbitrary associative ring, with the tilting module playing the role of the Gorenstein ring and the Cohen-Macaulay condition substituted by its homological characterization (see e.g. [6, Theorems 6.3, 12.3]).

References

[1] I. Assem. Tilting theory – an introduction, Topics in algebra, Part 1. Banach Center Publ. 26, Part 1. PWN, Warsaw 1990.

[2] S. Brenner and M. Butler. Generalizations of the Bernstein-Gelfand-Ponomariev re- flection functors, in “Proc. ICRA II, Ottawa 1979,” pp.103–169, LNM 832, Springer Verlag, Berlin, 1981.

[3] R. R. Colby and K. R. Fuller. Tilting, cotilting and serially tilted rings, Comm.

Algebra 18, (1990), 1585–1615.

[4] J. Herzog and E. Sbarra. Sequentially Cohen-Macaulay modules and local cohomology, preprint.

[5] Y. Miyashita. Tilting modules of finite projective dimension, Math. Z. 193, (1986), 113–146.

[6] R. P. Stanley. Combinatorics and Commutative Algebra, 2nd ed., Progress in Math- ematics 41, Boston, Birkh¨auser, 1996.

[7] A. Tonolo. Weakly cotilting modules of finite injective dimension, preprint.

(Alberto Tonolo) Dipartimento di Matematica Pura ed Applicata, Universit`a di Padova, via Belzoni 7, I-35131 Padova - Italy

E-mail address, Alberto Tonolo: tonolo@math.unipd.it

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