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COMMUNICATIONS IN ALGEBRA, 28(4), 2 147-2 172 (2000)

FINITELY COTILTING MODULES

Lidia Angeleri Hiigel ' Mathematisches Institut der Universitat

TheresienstraBe 39 D-80333 Miinchen

Abstract

Over an arbitrary ring, we consider cotilting modules endowed with some finiteness

conditions. We show that they correspond to pairs of dualities between certain cat-

egories consisting of finitely presented modules. This extends the Cotilting Theorem

proved by Colby for the noetherian case. The relationship with the cotilting modules

studied by Colpi, D'Este, Tonolo and Trlifaj is also discussed.

Introduction

Tilting theory was introduced in the early eighties in the context of finitely gen-

erated modules over artin algebras by Brenner-Butier [5] and Happel-Ringel

[19], and since then i t has played a central role in the development of the rep-

resentation theory of artin algebras. The dual of a tilting module with respect

to the usual artin algebra duality is called a cotilting module.

While the concept of a tilting module is now fairly well understood also

for infinitely generated modules over arbitrary rings [7], [17], the situation for

cotilting modules seems to be more complex.

Copyright Q 2000 by Marcel Dekker, Inc.

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2148 ANGELERI HUGEL

The first work aiming to a generalization of the cotilting modules considered

in the representation theory of artin algebras, was done by Colby in [S]. He

extended the notion of cotilting to modules which are finitely generated over

a noetherian ring and also noetherian over their endomorphism ring. For this

case, he could prove a Cotilting Theorem, which relates cotilting modules to

Morita duality just as Brenner's and Butler's Tilting Theorem had related

tilting modules to Morita equivalence.

Another generalization has then been studied by Colpi, D'Este, Tonolo and

Trlifaj in [14] and [16]. Their definition of a cotilting module is obtained by

dualizing the definition of a (possibly not finitely generated) tilting module

given in [17]. In recent work of Colpi and Fuller [13], [15] it is shown that

also this kind of cotilting theory extends Morita duality, and the relationship

to Colby's generalized Morita dualities [9] is clarified. Observe, however, that

even in the noetherian situation, this concept of cotilting is quite different from

the one investigated by Colby in his first paper [S] (see Examples 2.1 and 2.4).

As an attempt to find a bridge between these two approaches, we discuss

in this paper a notion of cotilting module endowed with certain finiteness

conditions. We call a left module RQ over an arbitrary ring R a finitely

cot i l t ing m o d u l e if it satisfies the following conditions:

(ii) Extk(Q, Q) = 0.

(iii) There is no nonzero module M such that Hom~(h1 , Q) = E x ~ R ( M , Q) = 0.

(iv) RQ is finitely generated, and the functor 4 = HomR( ,Q) : RMod + Mod (EndR Q)"P carries finitely generated modules to finitely generated

modules.

Our concept coincides with &!by's definition in [8] when we restrict our

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FINITELY COTILTING MODULES 2 149

attention to Morita rings (see Theorem 3.3). On the other hand, it coincides

with the definition given by Colpi, D'Este and Tonolo [14] in case the module

is finitely generated and product complete in the sense of [20] (see Proposition

2.2). In particular, all three notions coincide with the usual cotilting modules

when we are dealing with finitely generated modules over artin algebras.

We will see in section 1 that a finitely cotilting module satisfies quite strong

finiteness conditions. For instance, it is finitely presented over its endomor-

phism ring which, moreover, is left coherent. Let us point out that every tilting

(left) module, even every *-module, over a (left) noetherian ring satisfies the

dual finiteness conditions, and in particular also has a left coherent endomor-

phism ring. So, finitely cotilting modules can be viewed as the dual coun-

terparts of noetherian tilting modules, though they need not be noetherian,

neither over the ground ring nor over the endomorphism ring (see Example

3.6).

This is essentially the reason why Colby's Cotilting Theorem in [8] extends

to our setting. Indeed, our main result is a Cotilting Theorem for a notion of

cotilting which is even weaker than finitely cotilting. We extend Colby's defi-

nition to the non-noetherian case and say that a module RQ over an arbitrary

ring R is a Colby-module if it satisfies the foilowing coiiditiofis:

(i') Ext; (X, Q) = 0 for all finitely presented modules R X .

(ii) Exth (Q, Q) = 0.

(iii') There is no finitely presented nonzero module such that

H o m ~ (M, &) = Exth (bf, Q) = 0.

(iv) RQ is finitely generated, and the functor A = H o m ~ ( , Q ) : RMod --+

Mod (EndR Q ) " P carries finitely generated modules to finitely generated

modules.

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2150 ANGELERI H ~ ~ G E L

Further, we call a bimodule RQS a Colby-bimodule if it is a faithfully bal-

anced bimodule such that RQ and Qs are Colby-modules, or equivalently

(Proposition 4.3), if RQ is a finitely presented Colby-module over a left coher-

ent ring R and S ?Z (EndR Q)"P.

In case R is left noetherian and S is right noetherian, the Colby-bimodules

are precisely the modules considered by Colby in [8]. Observe however that R

being Ieft noetherian does not imply that S is right noetherian (Example 3.6).

Our Cotilting Theorem (Theorem 4.4) now asserts that a Colby-bimodule

RQS induces a torsion theory in the category of finitely generated R-, resp. S-

modules, and yields a pair of dualities, one between the torsion-free parts,

which consist of finitely presented modules, and one between the finitely pre-

sented modules in the torsion parts. We also extend Colby's notion of a fini-

tistic generalized Morita duality to arbitrary rings and see that the bimodules

defining such a duality are precisely the Colby-bimodules (Theorem 4.6).

In other words, it turns out that Colby's results in [8] still hold when we

replace the noetherian assumption by the above condition (iv). We would like

to remark that condition (iv) is rather natural. In fact, as we know from [2],

it just means that the category add Q is a covariantly finite subcategory of the

category Rmod of finiteiy generated moduies, a property shared by aii finitely

generated modules over an artin algebra.

Let us now fix some notation. For a ring R we denote by M o d R the

category of all and by m o d R the category of all finitely generated right

R-modules; the corresponding categories of left R-modules are denoted by

R M o d and R m o d . Given a class of modules M and a module X, we denote

by rM (X) the trace of M in X. Further, Add M (add M ) denotes the class

consisting of all summands of (finite) direct sums of elements of M . If M

consists just of one module M , then we write Add M, resp. add M.

For a module with S = (EndR Q ) O P , we denote dx = Hem ( , Q ) :

R Mod + Mod S and rR = E x t i ( , Q) : RMod + Mod S , the corresponding

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FINITELY COTlLTING MODULES 2151

functors defined by sQ are denoted by As and r s . When the context is

clear we simply write A and r. The kernel of will be denoted by 'Q.

Recall that a module X is said to be (A-)reflexive if the evaluation morphism

Jx: X -+ A2(X) given by dx(a) : x a ( x ) is an isomorphism. Of course,

since Ker dx coincides with the reject RejQ(X) of Q in X , all reflexivemodules

are in the category C o g e n Q of Q-cogenerated modules. In our context, an

important role will be played by the subcategories Y=Cogen Q n Rmod and

X=Ker A n Rmod of Rmod.

For details on tilting and cotilting theory, we refer to [22],[7].

1 Finiteness conditions

We start out with some properties of finitely cotilting modules.

P ropos i t ion 1.1 ([4, 5.1],[13, Lemma 4(a)]) Let RQ be a module and S =

end^ Q ) O P .

(1) Assume that RQ satisfies condition (ii) in the definition of finitely cotilting,

and let EX E CogenQ. Then A(X)s is finitely generated if and only if there

is an exact sequence 0 --+ X -+ Qn ---+ L --+ 0 for some n E IN and some

L E I Q .

(2) Assume that RQ satisfies conditions (i) and (iii) in the definition of finitely

cotilting. If ,QX and X/Rej9(X) are contained in 'Q, then X E CogenQ.

(3) Assume that Q is finitely cotilting. Then y = ' Q n Rmod.

For a cotilting module Q in the sense of [14] we know that every module

cogenerated by Q is copresented by Q (see (14, 1.81). For a finitely cotilting

module, we obtain the corresponding property restricted to finitely generated

modules. We say that X is finitely copresented by Q if there is an exact

sequence O -t X -+ Qn -i Qm for some n, m E IN.

Propos i t ion 1.2 Let RQ be a finitely cotilting module and S = c end^ Q ) O p .

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2152 ANGELERI HUGEL

Then the following statements hold true.

(1) Every module in y is finitely copresented by Q and reflexive.

(2) Every finitely generated module Ys E Cogen Qs is finitely presented. In

particular, Qs is finitely presented and S is right coherent.

(3) AR carries finitely generated modules to finitely presented modules.

(4) I?R carries finitely presented modules to finitely presented modules.

Proof: (1) The first statement is an immediate consequence of 1.1. Further,

any exact sequence 0 + X --+ Qn + L + 0 where n E IN, X E Y and

L E 'Q gives rise to the following commutative diagram

where Jx and SL are monomorphisms and bQn is an isomorphism. Then by

the snake lemma also Sx is an isomorphism.

(2) The claim is proven in [2] for any module RQ satisfying condition (iv) in

the definition of finitely cotilting.

(3) is an immediate consequence of (2).

(4) Let X be finitely presented. Then any exact sequence 0 + 1; -+ Rm -+ X -+ 0 with finitely generated I< yields an exact sequence A(Rm)s + A(I<), + r (X)S + 0 where the first two terms are finitely presented by

(3) . This shows that r(X)S is finitely presented.

0

In section 4 we will need the above Proposition in a slightly different version

in order to apply it on Colby-modules.

Remark 1.3 Statements (1)-(4) in Proposition 1.2 hold true also if RQ sat-

isfies condition (iv) in the definition of finitely cotilting and y consists of

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FINITELY COTILTING MODULES 2153

the finitely presented modules belonging to ' Q . The proof is the same as in

Proposition 1.2.

We now show that the dual finiteness conditions are satisfied for tilting

modules, or even *-modules, over a one-sided noetherian ring. Recall that

a right module MA over a ring A is said to be a *-module if it induces

an equivalence of categories in the sense of Menini-Orsatti's rtepresentation

Theorem [21]. This means that if we put S = EndMA and M* = HornA (11.1, W )

for some injective cogenerator WA, then the functors F = HornA (&I, ) :

ModA + M o d s and G = 8 s M : M o d s -t ModA induce mutually inverse

equivalences between the category Gen M of M-generated A-modules and the

category Cogen Ad* of M'-cogenerated S-modules. Moreover, MA is called a

(classical) tilting module if M is a finitely presented module satisfying (ii)

and the dual versions of conditions (i) and (iii) (see [17] and [ l l , Th. 3)). It is

shown in [ l l ] that MA is tilting if and only if it is a faithful finendo *-module.

Observe that by [24] every *-module MA is finitely generated. Hence

Gen M n m o d A , the dual counterpart of 7 , coincides with the category gen M

of the finitely M-generated modules. We say that a module X is finitely pre-

sented by 1Z.i i f there i b an exact seijiience Mm + 1'4" -+ X -+ D for sorr?e

n , m E IN.

Proposition 1.4 Let A be right noetherian and Ma a *-module with S =

End MA. Then the following statements hold true.

(1) Every module X E gen h.l is finitely presented by !ZI and satisfies G F ( X )

X .

(2) Every finitely generated module k;- E Cogenkl* is finitely presented. In

particular, S is a right coherent ring.

(3) F carries finitely generated modules to finitely presented modules.

(4) If h/lA is a (classical) tilting module, then F' = ~ x t a (&I, ) : ModA + M o d s carries finitely presented modules to finitely presented modules.

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2154 ANGELERI HUGEL

Proof: MA is finitely generated and hence noetherian. By 126, 27.31 we then

know that the category a[M] of all M-subgenerated modules is locally noethe-

rian. From [12, 3.2 and 6.11 it then follows that F ( X ) s is finitely generated

for all X E gen M . Since A is right noetherian, for every finitely generated

module XA we have TM(X) E gen M, and F ( X ) s F(TM(X))S is therefore

finitely generated. Now, we have proven in [2] that statement (2) holds true

for any noetherian module MA such that the functor F carries finitely gen-

erated modules to finitely generated modules. Further, statement (3) follows

immediately from (2).

Let us prove (1). Since MA is a *-module, we have G F ( X ) X for all

X E Gen M. So, the functor G maps a projective presentation of F ( X ) to an

Ad-presentation of X . Now, if X E gen M , then we know by (3) that F ( X ) s

is finitely presented. This yields an M-presentation &Im -+ Mn -+ X -+ 0.

It remains to prove (4). If MA is a (classical) tilting module, then it is well

known that F1(A)S is finitely ~resented. Indeed, we have an exact sequence

0 -+ A -+ M' --+ M" -+ 0 with MI, M" E add M (see [7],[17]), which

gives a finite projective presentation F(M1)s + F(M1')s -+ F1(A)s + 0

since ~ x t a (M,n/I) = 0. Moreover, using that p d M ~ 5 1, we have that any

exact sequence 0 -+ I< -+ Am -+ X -+ 0 yields an exact sequence

F'(I<)s + F'(Am)s + F'(X)s -+ 0. So, F' carries finitely generated modules

to finitely generated modules. Further, if X is finitely presented, then A' and

therefore also F 1 ( K ) are finitely generated, which shows that F1(X) is finitely

presented as well.

2 Cotiltings

Let us now compare finitely cotilting modules with other concepts of cotilt-

ing moduies. Recaii that cotiiting modules were introduced by Colpi, D'Este

and Tonolo in [14] by dualizing the definition of tilting modules given in [17].

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FINITELY COTILTING MODULES 2155

Namely, a module T is said to be t i l t ing if the category Gen T of T-generated

modules coincides with the kernel T of the functor Exth (T, ). This gener-

alizes the classical notion of a tilting module. Indeed, T is a classical tilting

module (see page 7) if and only if T is a tilting module in the above sense and

T is finitely presented [17, 1.1 and 1.31. Dually, in (141; a module Q is said to

be cot i l t ing if Cogen Q coincides with IQ.

Our definition of finitely cotilting differs from the one in [14] not only be-

cause of the finiteness conditions imposed on *Q in (iv), but also because

we require Ext;(Q1, Q) = 0 only for finite sets I. In fact, RQ is cotilt-

jng in the sense of [14] if and only if it satisfies (i), (iii) and the condition

(ii*) Exth(Q1, Q) = 0 for all sets I (see [14, 1.6 and 1.71).

E x a m p l e 2.1 (Trlifaj) An example of a finitely cotilting module which is

not cotilting in the sense of [14] is given by the regular module Z. In fact, zZL

obviously satisfies condition (i), (ii) and (iv) in the definition of finitely cotilt-

ing. Moreover, '22 consists of the Whitehead groups. Now, any Whitehead

group M is separable [18, Chapter XII, 1.31, so it has non-zero free summands

whenever M # 0 (see also 127, 3.41). Hence, Hornz (M, Z) = Ext&(M, 23) = 0

implies M = 0; and condition (iii) is satisfied. On the other hand, the group

22" is not slender and therefore not a Whitehead group [18, loc. cit.], which

shows that Exth (ZW,Z) # 0, and condition (ii*) fails.

However, the two notions coincide for finitely generated product complete

modules. According to (201, we call a module h.1 produc t complete if the

category Add 1V1 is closed for products.

P ropos i t ion 2.2 (1) A finitely generated product complete module RQ is

finitely cotilting if and only if RQ is a cotilting module.

(2) A module RQ of bite length is finitely cotilting if and only if RQ is a

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2156 ANGELERI HUGEL

cotilting module such that the functor AR carries finitely generated modules

to finitely generated modules.

Proof: (1) Let RQ be finitely cotilting and product complete. As in [15, 3.81,

we can now use the fact that every Q1 is isomorphic to a direct summand

of some Q(J) to conclude that Extfi (Q', Q) = 0 for any set I, and so Q is

cotilting. For the converse implication, observe that the functor AR carries

finitely generated modules to finitely generated modules whenever Q is product

complete. In fact, it follows from [20, 3.11 that every module X admits a map

f : X -k Q(') where I is a set and AR(f) is surjective, and of course, if X

is finitely generated, we can choose I to be a finite set, so that A R ( X ) ~ is a

factor of Sn for some n.

(2) By [20, 3.81, a finitely generated left module RQ is product complete if

and only if the ring S = (EndR Q)"P is right coherent and left perfect, and Qs

is finitely presented. Now, since the endomorphism ring of any finite length

module is semiprimary, we have that S is left perfect. Moreover, we have

shown in [2] that S is right coherent and Qs is finitely presented whenever RQ

satisfies condition (iv) in the definition of finitely cotilting. So, our statement

follows from (1).

The following example shows that we cannot omit the condition on AR in

the above statement (2).

Example 2.3 Let F C G be a skew-field extension such that dimFG = 2

and dimGF = cu (see [ 6 ] ) . Then the triangular matrix ring R = [: Y ) is a left artinian hereditary ring, and P = R e l is a cotilting module of fi-

nite length which is not finitely cotilting. In fact, since P is not finitely

generated over F = (EndR P)"P, condition (iv) in the definition of finitely

cotilting fails. On the other hand, we can use arguments similar to those

in [14, Example 5.31 to see that P is cotilting. In fact, id(P) 5 1 since R

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FINITELY COTlLTlNG MODULES 2157

is hereditary. Moreover, by Chase's theorem, Pi is projective and therefore

E x t i ( P i , P ) = 0 for any set I. So, (i) and (ii*) are satisfied, and it re-

mains to verify (iii). Assume to the contrary that there is a nonzero module

M with H o m ~ (M, P) = Extfi ( M , P ) = 0. Observe that since Rez c P,

we also have HornR (M, Re2) = 0. We then deduce that M has a finitely

generated indecomposable non-projective submodule X , because otherwise we

could write M as a direct limit of projective modules, which would imply

that M is flat and hence, by Bass' theorem, n/l would be projective. Now,

since R is hereditary, HomR (X, P ) = Extfi (X, P ) = 0, and so X is a finitely

P-generated module with an exact sequence 0 4 fi' + Pn + -Y 4 0

where K G Re2 " and m, n E IN. Applying H o m ~ ( , P), we obtain an F-

isomorphism HornR (Pn, P ) '2 H o m ~ (I<, P ) , which yields Fn F Gm F , a

contradiction.

We now turn to the cotilting modules studied by Colby in [S]. Colby as-

sumed that RQ is a finitely generated left module over a left noetherian ring

R and also a finitely generated right module over the right noetherian ring

S = (EndR Q ) " P , and introduced a concept of cotilting for this setting. In the

iiitiodiictioii, we have extended his definition to the non-noetherian case by

means of the notion of a Colhy-module. Let us point out that every finitely

cotilting module is a Colhy-module, while the converse is not true, not even

in the noetherian case.

Example 2.4 (TI-lifaj) Let p be a prime number and J, the ring of all padic

integers. Of course, the regular module J, satisfies conditions (i), (ii), and

(iv) in the definition of finitely cotilting. By the structure theory of finitely

generated modules over a principal ideal domain, it also satisfies condition

(iii*). So, j,JP is a Colby-module.

On the other hand, since Jp is a complete discrete valuation ring, we know

from [18, Chapter XII, 1.171 that any torsion-free Jp-module, so in particular

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2158 ANGELERI HUGEL

&,, is a Whitehead-module, that is ~ x t : ~ (Q,, J,) = 0. Moreover, since Q,

is divisible and J, is reduced, we have Hom J, (&,, J,) = 0. Thjs shows that

condition (iii) fails, and so j,Jp is not finitely cotilting.

The following results will be useful in the sequel.

Lemma 2.5 Let RQ be a module such that all R-modules in y are finitely

presented and belong to iQ. Then for any finitely generated module R M we

have h/fl = RejQ(kf) E X and M/M1 E y with A(12f) g A(M/M1). Moreover,

l?(M) r F(M1) if RQ satisfies the above condition (i'), and in particular, if

RQ is faithful.

Proof: Consider the exact sequence 0 + M' + M -fi M/M1 -+ 0.

Since M/M1 E CogenQ n mod R = y is finitely presented, we have that

M' is finitely generated. Moreover, we have a long exact sequence 0 -t

A(M/M1) A2 A(M) -t A(M1) -t r(M/M1) = 0 + r ( M ) -t r(M1) -t

E x t i (M/M1, Q) where A(") is an isomorphism by construction. Hence

h ( M 1 ) = 0, and so M' E I<er A n Rmod = X.

For the additional statement, it suffices to verify that Ext;(M/M1,Q) = 0.

This is clear in case Q satisfies (i*). We now show that Q satisfies (i')

whenever it is a faithful module. Let X be a finitely presented module and

0 --+ I< + Rn + X -+ 0 an exact sequence with finitely generated I<.

If R E CogenQ, we have that Rn and h' are both contained in y, and in

particular, I< E 'Q. We then conclude that l?(I<) = 0 = Ext;(Rn, Q), thus

~ x t ; ( X , Q) = 0.

Proposition 2.6 The following statements are equivalent for a module RQ.

(1) R is left coherent and RQ is a finitely presented Colby-module.

(2) RQ satisfies condition (iv) in the definition of finitely cotilting, and y

consists of the finitely presented modules belonging to 'Q.

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FINITELY COTILTING MODULES

Proof: (1)+(2) : We follow the same line as in the proof of [14, 1.71 and of

1.1(3). Let X E y. Then we know from 1.1 (1) that there is an exact sequence

0 --+ X --+ Qn -+ L --+ 0 for some n E IN and some L E ' Q . Since Qn

is coherent, both X and L are finitely presented. Moreover, condition ( i* )

and (ii) imply l?(Qn) = 0 = Ext i (L ,Q) , thus r ( X ) = 0. This shows that all

modules in y are finitely presented and belong to 'Q.

Let us now take a finitely presented module X E 'Q and denote X' =

RejQ(X). By Lemma 2.5 we know that X' is a finitely generated module

with AR(X') = 0 and r ( X i ) n r ( X ) = 0. Moreover, since X is coherent, the

submodule X' is even finitely presented. This implies X' = 0 by condition

(iii*) and proves that X E. Cogen Q.

(2)+(1) : RQ, R and all finitely generated left ideals belong to y . This shows

that R is left coherent and that RQ is finitely presented and faithful. We then

know from the proof of 2.5 that condition (i') is satisfied; (ii) and (iii') are

left to the reader.

3 Morita rings

V!e are now going to see that over Morita rings the finitely cotilting modules

coincide with the Colby-modules and arise in a natural way as duals of tilting

modules. Recall that a ring R is said to be left M o r i t a if there is a Morita

duality D : Rmod +, mod.4 for some ring A, or equivalently, if R is left

artinian and the minimal injective cogenerator EiW is finitely generated.

First of all, we discuss Morita duals of tilting modules. Let TA be a classical

tilting module with S = EndTA, and assume that there is a faithfully balanced

bimodule RWA which is an injective cogenerator on both sides and therefore

induces a Morita duality. In [ l j , $21, the bimodule RQS = H O ~ A ( T , kv) was investigated, and it was shown that RQS is faithfully balanced, Qs is

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2160 ANGELERI HUGEL

a cotilting module in the sense of [14], i. e. CogenQs = Qs, and RQ is a

finitely cogenerated module with following properties:

(i) WRQ) I 1.

(ii) Extk(Q, Q) = 0.

(iii*) There is an exact sequence 0 --+ Q1 --+ Qo + R W --+ 0 where

Qo1 Q1 E add Q.

Here, we want to study how the finiteness conditions on tilting modules

considered in Proposition 1.4 behave under Morita duality.

Lemma 3.1 (1) RQ has finite length if and only if TA has.

(2) Assume that A is right noetherian. If R X is a finitely generated module

such that the A-module HornR ( X I W)* is finitely generated, then

H o m ~ (X, Q)s is a finitely presented S-module. In particular, Q s is finitely

presented if and only if WA is finitely generated.

Proof: (1) By [7, l.l],we have that A r EndsT. So, we can interpret

Q s = H o m ~ (sTA, WA) = (ST)+ as the local dual of ST, and by [23, Th. 21

we know that S T is endofinite if and only if Qs is endofinite. This gives the

claim, since we also have that R 2 Ends Q (by [15, 2.31).

(2) For such A and R X we have HornR (XI Q)s H O ~ A (TI H O ~ R (XI W)A)S

(see for instance [15,2.l(b)]), and we know from Proposition 1.4 that the latter

module is finitely presented over S .

In particular, if we take X = R, then we have that Q s is finitely presented

whenever PVA is finitely generated. For the converse implication, recall that the

injective module WA lies in GenT and therefore WA H O ~ A (T, W) 63s T =

Qs @J T . So, if Qs is finitely presented, then WA is finitely presented. 0

Let us now focus on Morita rings.

Lemma 3.2 Let R be a left artinian ring admitting a finitely generated in-

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FINITELY COTILTlNG MODULES 2161

jective cogenerator RW, and let RQ be a finitely generated module satisfying

conditions (i*), (ii), (iii*) in the definition of Colby-module. Then the following

statements are equivalent.

(i') I'(W)s is finitely generated.

(ii') l? carries finitely generated modules to finitely generated modules.

(iii*) There is an exact sequence 0 + Q1 -+ Qo -+R W -+ 0 where

Qo,&I E addQ.

(iv') A carries finitely generated modules to finitely generated modules.

Proof: (iii*)+(il) : The given exact sequence yields an epimorphism

A(Ql)s -+ J?(W)S + r(Q0) = 0 where A(Q1) is finitely generated projec-

tive.

(i')+(ii') : Let RX be finitely generated, hence finitely cogenerated with an

exact sequence 0 -+ X -+ W n -+ L --+ 0. Then L is finitely presented,

hence E x ~ ~ ( L , Q ) = 0 by (i*). SO, J?(Wn)S -+ II(X)s + 0 is exact, thus

II(X)s is finitely generated.

(ii')*(iv'): Let R X be finitely generated, and let X ' = RejQ(X). Then

A ( X ) s 2 A(X/X')s, and X/X1 is artinian and Q-cogenerated, hence finitely

cogenerated by Q with an exact sequence 0 --+ XJX' --+ Qn -+ L -+ 0.

So, we obtain an exact sequence Sn + A(X/X')s -+ I'(L)s -+ r(Qn) = 0.

Since RL, and therefore also I'(L)s, are finitely generated, we conclude that

A(X/X')s is finitely generated.

(iv')=+(iiiS) : Observe first that Q is a Colby-module, hence y consists of the

finitely presented modules belonging to ' Q by Proposition 2.6. The proof now

works as in [3, 2.31, but all involved cardinalities are here finite. Indeed, R W

is finitely generated, say by Rn, and Rn E y is finitely cogenerated by Q.

Moreover, the kernel of any map f : Qm -+ W is finitely Q-copresented by

1.3. 0

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2162 ANGELERI HUGEL

Theorem 3.3 Let R be a left Morita ring, and assume that the bimodule

RWA induces a Morita duality D = Hom( , W) : Rmod ++ modA. Then the

following statements are equivalent for a left module RQ.

(1) There is a classical tilting module TA such that RQ D(TA).

(2) R Q is a finitely generated module with the above properties (i), (ii), (iii*).

(3) RQ is a Colby-module.

(4) R Q is finitely cotilting.

Under any of these conditions we also have that Qs is a finitely presented

finitely cotilting module.

Proof: First of all, recall that our assumption implies that R is left artinian,

A is right artinian, and RWA is finitely generated on both sides. From 3.1

we then obtain that Qs is finitely presented under condition ( I ) , and further,

that RQ has finite length under any of the conditions (1) - (4). The latter

implies that the functor As = R HornS ( , Q) : M o d s -+ RMod carries finitely

generated modules to finitely generated modules. So, since we know from [15,

2.41 that Qs is cotilting, we conclude that Qs is also finitely cotilting. Let us

now prove the equivalence of the stated conditions.

(1)+(2) : It was proven in [15, 2.51 that RQ is a finitely cogenerated module

satisfying (i), (ii), (iii*). Since RW is noetherian, RQ is also finitely generated.

(2)*(1) : Applying D, we obtain that T = DQA is a finitely presented module

satisfying condition (ii) and the dual versions of (i) and (iii*), hence a classical

tilting module. Of course, DT E RQ.

(2)+(4) : Observe first that any module satisfying (2) has property (iii) in

the definition of finitely cotilting. Indeed, take a module R M such that

HornR (M, Q) = Extk(M, Q) = 0. Applying the functor H o m ~ ( M , ) on the

sequence in (iii*), we see that H o m ~ (M, W) = 0, hence M = 0. The claim

now follows from Lemma 3.2.

(4)+(3) : is obvious.

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FINITELY COTILTlNG MODULES 2163

(3)+(2) : Since R is left artinian, condition (i*) means that E x t i (X, Q) = 0

for all finitely generated modules RX, which is equivalent to i d ( ~ Q ) < 1 by

Baer's criterion. So, RQ is a finitely generated module with properties (i), (ii),

(iii*), (iv'), and by Lemma 3.2 it has also property (iii*).

Remark 3.4 Combining Theorem 3.3 with Proposition 2.2, we obtain that

all cotilting notions discussed so far coincide for finitely generated modules

over artin algebras.

We point out that in the hereditary case the connections are even closer.

P r o p o s i t i o n 3.5 Let R be a hereditary left Morita ring. Then the following

statements are equivalent for a left module RQ.

( I ) RQ is finitely cotilting.

(2) RQ is a cotilting module with R& and Q(EndRQ)oP being finitely generated.

(3) ,QQ is a classical tilting module.

P r o o f : (1)*(2): This holds over any left artinian ring by Propositions 2.2(2)

and 1.2(2).

(2)+(3): This is shown in 14, 5.31 for any left artinian hereditary ring.

(3)+(1): If R is a left hereditary left noetherian ring, then we know from [7,

2.11 that (3) implies conditions ( i ) , (ii), and (iii'). We will now verify (iii*)

under the additional assumption that the minimal injective cogenerator

is finitely generated. In fact, in this case R W E QL = GenQ is finitely Q-

presented by Proposition 1.4 and therefore admits an exact sequence 0 -+ A' --+ Qn --+ W 4 0 where Ii E gen Q. So, our proof will be complete

once we have shown that I< E add Q. Note that also I< is finitely Q-presented

and admits an exact sequence 0 --+ L --+ Qm -+ Ii' -+ 0 where L E gen Q.

But then Extk (Qn, L) = 0, and further, EX^; (W, L) = 0 since R is hereditary.

Hence Extk (11') L) = 0, and the latter sequence is even split exact, which

implies K E add Q. 0

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2164 ANGELERI HUGEL

We now show that Colby-modules over a left noetherian ring need not be

noetherian over their endomorphism ring.

Example 3.6 By an example of Tachikawa and Valenta [25, $11, there exists a

right artinian (hereditary) ring A such that the minimal injective cogenerator

is a classical tilting module TA and S = End TA is left but not right noetherian.

Then, in the above notation, WA = TA, S % R, and RQS = Homa(T,T) 2

sSs is a faithfully balanced bimodule such that RQ and Qs are finitely cotilting

by Theorem 3.3, and RQ is noetherian, but Qs is not noetherian.

4 A cotilting theorem

This section is devoted to a generalization of Colby's work in [S]. Let us first

briefly summarize his results. Under the general assumption that R is left

noetherian and S is right noetherian, Colby has shown that a module RQ is

a Colby-module with S = (EndR Q)OP if and only if RQS is a Colby-bimodule,

that is, a faithfully balanced bimodule such that both RQ and Qs are Colby-

modules [8, 3.31. Moreover, he has proven that a Colby-bimodule induces

a pair of dualities A : y H y and r : X +, X such that I'A and A r map

any finitely generated module to zero [8, 2.41. Finally, he has characterized the

Colby-bimodules in terms of some duality conditions which generalize the case

of Morita duality and which he termed finitistic generalized Morita duality 18,

2.31.

It will now turn out that one does not need the full power of the noetherian

assumption for these results. In fact, roughly speaking, the key point in Colby's

work is just the fact that the modules satisfy condition (iv) in the definition of

finitely cotilting, which, as we know from Example 3.6, is a weaker assumption.

We start out with some preliminary results.

Lemma 4.1 Let RQ be a module with S = (EndRQ)OP. Assume that RQ is

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FINITELY COTILTING MODULES 2165

faithful, and that all modules in y are reflexive. Then the following statements

hold true for every finitely presented module X E X.

(1) r R ( X ) E Ker As.

(2) If rR(X) = 0, then X = 0.

(3) If ~ S A R maps any finitely generated module to zero, then X % r s r ~ ( X ) .

P r o o f : Let X E X be finitely presented with an exact sequence 0 + h' '. R" -% X + 0 where h' is finitely generated. Then we obtain a

commutative diagram

Observe that Rn, and therefore also its finitely generated submodule h', belong

to y. Thus dK and JRn are isomorphisms, and AsAR(i) is a monomorphism.

So, AsI?R(X) = 0, showing (1). Further, if rR(X) = 0, then As&(i), and

therefore also i, are isomorphisms, which proves (2). Finally, if I's&(I<) = 0,

then yx must be an isomorphism, and we obtain (3) . 0

Corol lary 4.2 Let RQ be a module which satisfies condition (iv) in the def-

inition of finitely cotilting. Assume further that RQ is faithful and that all

modules in y are finitely presented, reflexive and belong to 'Q. Then R is

left coherent and RQ is a finitely presented Colby-module.

P r o o f : Like in the proof of 2.6 we obtain that R is left coherent and RQ

is finitely presented. Let now M be a finitely presented module in 'Q. Then

M is coherent, hence we know by Lemma 2.5 that M' = Rejg(M) is a finitely

presented module in X with rR(M1) = 0. By Lemma 4.1 we then conclude

that M' = 0, hence M E y. So, we have shown that Y consists of the finitely

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presented modules belonging to I Q , which means by Proposition 2.6 that RQ

is a Colby-module. 0

The following Proposition extends Colby's [8, 3.31.

Proposition 4.3 The following statements are equivalent for a bimodule RQS.

(1) RQS is a Colby-bimodule.

(2) R is left coherent, RQ is afinitely presented Colby-module, S end^ Q)OP.

Proof: (1)=+(2) : Since Qs satisfies condition (iv) in the definition of finitely

cotilting, we obtain as in 1.2(2) that R is left coherent and RQ is finitely pre-

sented.

(2)+(1) : By Proposition 2.6 we have that y = yR consists of the finitely pre-

sented modules belonging to I Q . Combining this with Remark 1.3, we obtain

that all modules in yR are reflexive, and further, that Q s and all modules

in Ys = Cogen Qs n mod S are finitely presented over the right coherent ring

S . In particular, we obtain that the regular module R is reflexive and RQS is

therefore faithfully balanced.

We want to apply Corollary 4.2 on Qs. In order to verify the assumptions

on Qs , we will use arguments similar to those employed by Colby in [8, 511.

Let us start by showing that all modules in ys are reflexive. To this end,

it suffices to show that if N 6 mods , then the evaluation morphism 6~ is

an epimorphism. Take an exact sequence 0 -+ h' -+ Sn --% N --+ 0.

Then RL = Coker As(g) c As(K) E Cogen Q is finitely generated as a fac-

tor of As(Sn) 2 RQn, hence lies in yR c ' -Q. Thus we have a commutative

diagram

O - + I i - + Sn + N -+0

4 6sn 4. $ 6 ~

0 -+AR(L) -+ARAs(S") ---+ARAs(N) -+0

where dSn is an isomorphism, which proves our claim.

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FINITELY COTlLTlNG MODULES 2167

Next, we show that all N E ys belong to '-Q. Note first that R M = As(N) is

reflexive. Moreover, since A R ( M ) ~ = ARAs(N) %! Ns is finitely generated, we

know from 1.1(1) that there is an exact sequence 0 ---+ M --+ Qn -% L --+ 0

for some n E IN and some L E ' Q . But then we obtain a commutative

diagram

where 6b1 and dQn are isomorphisms, hence dL is a monomorphism by the snake

lemma. This shows that L belongs to yR and is therefore reflexive. So, dL

is even an isomorphism, and AsAR(g) is an epimorphism. We conclude that

~ S ( A R ( M ) ) = 0 and Ns 2 AR(M)s E 'Q.

It remains to show that As carries finitely generated modules to finitely gen-

erated modules. As we have shown in [2], this follows from the fact that RQ

and all modules in yR are finitely presented.

Now we conclude from Corollary 4.2 that Qs is a Colby-module.

We are now in a position to prove a Cotilting Theorem which extends

Colby's [8, 2.41. For a module RQ with S = (EndR Q ) O P , we will consider as

usual the subcategories y and X of Rmod, and we will denote by y' and X'

the subcategories of y, resp. X , consisting of the finitely presented modules.

For sake of simplicity, the corresponding subcategories of m o d s will be also

denoted by y, X and y', XI, and we will write A and r both for AR and As,

resp. r~ and rs.

T h e o r e m 4.4 (Cotil t ing Theorem) The following statements are true for

a Colby-bimodule RQS.

(a) ( X , y ) is a torsion theory in Rmod resp. mods.

(b) A : y' +, y' and I? : X' +t X' are dualities, and y = 3''.

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2168 ANGELERI HUGEL

(c) FA maps any finitely generated module to zero, and A F maps any finitely

presented module to zero.

Proof: Combining Proposition 4.3 and 2.6, we know that y consists of the

finitely presented modules belonging to LQ. SO, we can apply Lemma 2.5, and

further, we know from Remark 1.3 that all modules in y are reflexive, and

that I? carries finitely presented modules to finitely presented modules. Let us

now prove the stated properties.

(a) It is easily checked that a finitely generated module M satisfies

H o m ~ (kf, Y) = 0 for all Y E y if and only if it lies in X. Assume now

HomR(X, M ) = 0 for all X 6 X. Then by Lemma 2.5, the submodule

M' = Rejq(M) E X must be zero, hence M E y.

(b), (c) If Y is finitely generated reflexive, then so is A(Y). Hence A in-

duces a duality y + y. This implies that FA maps any finitely generated

module to zero. In fact, if M is finitely generated, then M/M1 6 y , and so

A(M) E A(M/M1) E Ker J?. That y = y' has already been remarked above.

Let now X E X be finitely presented. Then r ( X ) is finitely presented, and

it is contained in X with F2(X) 2 X by Lemma 4.1. We conclude that I?

induces a duality X' -+ XI. Take now a finitely presented module M. Since

R and S are left, resp. right, coherent by Proposition 4.3, the module M is

coherent. So, we know from 2.5 that M' E X' with r ( l l I ) 2 f'(M1) E X', thus

AF(M) = 0. R

Remark 4.5 Of course, Theorem 4.4 holds in particular if RQS is a faithfully

balanced bimodule with RQ and Qs finitely cotilting modules. In this case we

have that X = X' if and only if I' carries finitely generated modules to finitely

generated modules, which is further equivalent to R being left noetherian,

resp. S being right noetherian.

Proof: If X = X', then for all finitely generated modules M we have

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FINITELY COTILTING MODULES 2169

M' E X', and r ( M ) Z r ( M 1 ) E X' is even finitely presented. Assume now

that r carries finitely generated modules to finitely generated modules, and

consider a left ideal A C R. The exact sequence 0 + A --+ R + h' -+ 0

yields an exact sequence A(R) -t A(A) + r ( K ) + 0 where A(R) and ~ ( I I ' )

are finitely generated. Thus A is a module in CogenQ such that A(A) is

finitely generated, and by Proposition 1.1(1) there is an exact sequence 0 + A --+ Qn -+ L ---t 0 for some n E IN and some L E 'Q. Observe

that RQ being finitely cotilting now implies y = ' Q f l Rrnod by 1.1(3), and

moreover, we have as in the proof of Theorem 4.4 that all modules in y are

finitely presented. So, the finitely generated module L lies in y and is finitely

presented. This shows that A is finitely generated, and so we have proven that

R is left noetherian. Finally, since R being left noetherian implies X = X',

our proof is complete. 0

Extending Colby's definition in [8, 511, we will say that a bimodule R Q ~

over two arbitrary rings R and S defines a finitistic generalized Mor i t a

duality in case

(I) AR and As carry finitely generated modules to finitely generated modules,

and R R and Ss are reflexive.

(11) Every finitely generated submodule of a finitely generated reflexive mod-

ule is reflexive.

(111) Any extension of a finitely generated reflexive module by a finitely gen-

erated reflexive module is reflexive.

Again, this definition coincides with the one given by Colby if R is left

noetherian and S is right noetherian. Compared with the usual Morita duality,

we restrict our attention to finitely generated modules, ensure that A behaves

well on them, and give up one closure property of the reflexives, namely the

closure under factors.

We will now characterize the bimodules which define a finitistic generalized

Morita duality. This extends Colby's [8, 2.31.

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2170 ANGELERI HUGEL

Theorem 4.6 The following statements are equivalent for a bimodule

(1) RQS is a Colby-bimodule.

(2) RQS is a faithfully balanced bimodule with properties (a)-(c) in 4.4.

(3) RQS defines a finitistic generalized Morita duality.

Proof: (l)+(2) : This is just Theorem 4.4.

(2)=+(3) : First of all, note that R R and Ss are reflexive if and only if RQS

is a faithfully balanced bimodule (see (1, 20.161). Moreover, as an immediate

consequence of (b), we have that A carries finitely generated modules to finitely

generated modules, so property (I) is verified. We then deduce that RQ and

Qs are finitely generated. Next, we infer from (b) and (c) that all modules

in y are finitely presented, reflexive and belong to 'Q. As in the proof of

Corollary 4.2, we conclude that y consists of the finitely presented modules

belonging-to 'Q. Hence y is closed under extensions, and property (111) is

verified. Note further that y is closed under finitely generated submodules by

condition (a), which proves property (11).

(3)+(1): Property (I) implies as in 1.2(2) that RQ, QS and all modules in

y are finitely presented. Moreover, by the same arguments as in [8, 1.11, we

deduce from (I) and (11) that A is exact on short exact sequences with all

modules finitely generated reflexive, and use (111) to conclude that all finitely

generated reflexives belong to 'Q. In particular, ~ x t k ( Q , Q ) = 0, and by

1.1(1), every module Y E y is finitely cogenerated by Q and therefore reflexive

by (11). So, we have verified the assumptions of Corollary 4.2, and conclude

that RQ and Qs are Colby-modules.

Acknowledgements: The main results of this paper were presented at

the Workshop on Cotilting Theory held at the University of Munich in March

1998. I am indepted to Riccardo Colpi, Gabriella D'Este, Enrico Gregorio,

A!herto Tono!n and Jan Tr!ifaj for many fruitful discussions on the subject.

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FINITELY COTILTING MODULES 2171

In particular, I wish to thank Jan Trlifaj for having contributed the examples

2.1 and 2.4. I also acknowledge a HSPIII-grant of the University of Munich

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[26] R. WISBAUER: Grundlagen der Modul- und Ringtheorie, R. Fischer Miinchen 1988.

[27] B . ZIMMERMANN-HUISGEN, Pure submodules of direct products of free modules, Math. Ann. 224 (1976), 233-245.

Received: March 1999

Revised: June 1999

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