Constructing Large Modules over Artin Algebras

Ž . 

JOURNAL OF ALGEBRA 187, 413421 1997 

ARTICLE NO. JA966832 

Constructing Large Modules over Artin Algebras 

Henning Krause 

Fakultat ¨ fur ¨ Mathematik, Uniersitat ¨ Bielefeld, 33501 Bielefeld, Germany 

Communicated by Gernot Stroth 

Received October 12, 1995 

Let be an artin algebra and denote by modŽ . 

the category of finitely 

generated -modules. Apart from those we are also interested in -modules 

which are not finitely generated. These are called large. Recall that 

the algebra is of infinite representation type, provided that modŽ . 

has 

infinitely many isomorphism classes of indecomposable objects. A theorem 

of Auslander asserts that there exists a large indecomposable -module 

whenever is of infinite representation type 2 . In this note we use his 

method to construct specific large indecomposable modules which arise 

naturally from certain infinite families of morphisms in modŽ . 

having 

isomorphic kernels. We apply this result as follows. Suppose there is given 

a chain 

 

1 2 s1 s 

X1 X2 Xs 

 

 

 

 

 

of monomorphisms between finitely generated indecomposable -modules. 

Denote by X 

lim Xi 

the corresponding direct limit. It is natural 

to ask for conditions that ensure the indecomposability of X . In fact, 

chains of monomorphisms of the above form occur for quasi-serial components 

of the AuslanderReiten quiver of . Ringel calls any module 

X Xs 

belonging to such a component quasi-serial of quasi-length s, 

since it determines, up to isomorphism, uniquely a chain of irreducible 

monomorphisms i: Xi X i1, i , such that there exists no irre- 

ducible monomorphism ending in X 

1 9 . We show that the corresponding 

direct limit X 

lim Xi 

is a large indecomposable module. Moreover, we 

obtain an exact sequence 

 

 

 

X 

s 

0 X X Tr D X 0, 

 

 

 

 

413Ž . Ž . 

 

0021-869397 $25.00 

Copyright 1997 by Academic Press 

All rights of reproduction in any form reserved.

414 

HENNING KRAUSE 

where Ž Tr D. s Ž X. 

denotes the sth power of the transpose of the dual of X 

which is again a quasi-serial module. This exact sequence has the following 

property. Every morphism : X Y in modŽ . 

belongs to the infinite 

radical of modŽ . 

if and only if factors through X , and moreover, 

induces a commutative diagram of the form 

 

X 

 

Ž . 

sŽ . 

 

0 X X Tr D X 0 

 

 

 

 

 

 

 

0 Y 

Y 

Y Ž . 

t 

 

Ž . 

 

 

 

 

Tr D Y 0 

provided that Y is quasi-serial of quasi-length t. 

We would like to point out as an application that modules of the form 

X 

can be used to construct generic modules in the sense of Crawley- 

Boevey 5 ; we refer the reader to 8 . Note that the existence of a generic 

module is crucial for the validity of the second BrauerThrall conjecture 

for artin algebras 5 . 

 

THE CONSTRUCTION 

Let be an artin algebra and denote by ModŽ . 

the category of right 

-modules. Given two morphism : X Y , i 1, 2, in ModŽ . 

i 

i 

we 

define 2 1 if there exists a morphism : Y1 Y2 such that 2 

1. We write 2 1 if 2 1 and 1 2. Recall that a morphism 

: XY in modŽ . 

is irreducible, provided that is neither a split 

monomorphism nor a split epimorphism and if for any factorization 

in modŽ . 

2 1 either 1 is a split monomorphism or 2 

is a split 

epimorphism. 

 

i 

 

i 

THEOREM 1. Let 0 X Yi Zi 0, i , be a family of exact 

sequences in modŽ .. Suppose that i is irreducible and that i1 i 

for 

all i . Then there exists a family of morphisms i: Yi Yi1 

satisfying 

i1 ii for all i such that Y lim Yi 

is a large denumerably 

generated indecomposable module. 

We postpone a proof and discuss some preliminaries. We shall use the 

category ŽmodŽ ., Ab. of additive functors from modŽ . 

to the category 

Ab of abelian groups. Note that for a pair of morphisms i: X Y i, 

i 1, 2, in modŽ . we have iff ImŽHom Ž , .. 

2 1 2 

ImŽHom Ž , .. as subfunctors of Hom Ž X, . 

1 . We shall also use the fully 

faithful functor 

 

ModŽ . Ž modŽ op .,Ab., XX 

,

LARGE MODULES 415 

which identifies the pure-injective -modules with the injective objects in 

Ž Ž 

op 

mod ., Ab .. Note that Ext 1 

Ž , X . 

vanishes on finitely presented 

functors for every -module X. Moreover, every finitely generated - 

module is automatically pure-injective. The following characterization of 

an irreducible morphism in modŽ . 

is needed. 

LEMMA 2. For a nonsplit exact sequence 0 X Y Z 0 in 

modŽ . 

the following are equialent: 

Ž. 1 is irreducible. 

Ž. 2 or for each morphism : X Y in modŽ .. 

Ž. 3 or for each morphism : X Y in ModŽ .. 

Proof. Ž. 1 Ž. 2 See 3, Proposition 2.7 . 

Ž. 23 Ž. One direction is trivial. Therefore suppose Ž. 2 . We apply the 

Ž Ž . . Ž Ž 

op 

well-known duality fp mod ,Ab fp mod ., Ab. 

between the categories 

of finitely presented functors which sends Hom Ž X, . 

to X 

Že.g., see . 6 . It follows from this duality that Ž 2. 

is equivalent to the 

Ž Ž 

op 

property that any subobject G of X in mod ., Ab. 

 

either contains 

or is contained in F KerŽ .. Taking G KerŽ . 

 

for a 

morphism : X Y we obtain either a morphism Y 

 

Y if F G since Ext 1 

Ž Z , Y . 

0, or we obtain a 

morphism Y Y if G F since Ext 1 

Ž , Y . 

0. Thus 

we have or . 

Let C be an abelian category and F: C Ab be an additive functor. A 

nonzero element x FŽ X. 

is called minimal if for each : X Y in C 

the property FŽ .Ž x. 

0 implies that is a monomorphism. The following 

facts are easily verified. 

LEMMA 3. Ž. 1 If x FŽ X . is minimal, then X is indecomposable. 

Ž. 2 If X is a noetherian object in C and x FŽ X . is a nonzero element, 

then there exists an epimorphism : X Y such that FŽ .Ž x. FY Ž . is 

minimal. 

Proof. See 1, Lemma 3.2 . 

Proof of Theorem 1. Let F Hom Ž X, . ImŽHom Ž , .. 

i i and 

denote by : Hom Ž X, . F the canonical morphism in ŽmodŽ .,Ab. 

 

. 

Using the preceding lemmas one constructs easily a family of morphisms 

i: X W i, i , together with a family of natural numbers 1 n0 n1

416 


n such that 

2 

Ž. i Ž . FW Ž . is minimal for all i ; 

i 

i 

Ž ii. 

for all i . 

ni 

i ni1 

Ž . 

Applying ii we obtain the following chain of morphisms. 

 

1 n12 n1 n22 

1 n11 1 n1 n21 2 

Y Y W Y Y W 

 

 

n2 

 

Y . 

n 2 

 

 

 

 

 

 

Taking the corresponding compositions in this chain we obtain morphisms 

: Y Y and : W W , i , such that 

i i i1 i i i1 

Ž iii. 

i1 ii and i1 ii 

for all i ; 

Ž iv. 

Y lim Y and W lim W are isomorphic. 

i 

 

i 

 

To show that Y is indecomposable we extend the functor F: modŽ . 

Ab 

to a functor F: ModŽ . 

Ab which commutes with direct limits and 

denote by : Hom Ž X, . 

F the corresponding morphism between 

functors from ModŽ . to Ab. Using Ž i. 

it is not hard to check that the 

morphism lim has the property that Ž . FŽ W . 

i 

is minimal. 

We refer to the proof of 2, Theorem 1.5 

for details and conclude that Y 

is indecomposable. 

 

Our next aim is to discuss some properties of the morphism 

lim i: X Y which arises in Theorem 1. We will use the following 

general fact. 

 

PROPOSITION 4. Let i: X Y i, i I, be a directed family of morphisms 

in ModŽ . 

and denote by lim i: X Y its direct limit. Suppose also 

that Y ModŽ . 

is pure-injectie. Then a morphism : X Y factors 

through if and only if factors through i 

for all i I. 

 

Proof. One direction is clear. Therefore suppose that factors through 

i 

for all i I. We obtain for each i an exact commutative diagram 

0 F X Y 

 

 

 

i i 

 

0 G X Y


and therefore also an exact commutative diagram 

0 limF X Y 

 

 

 

 

i 

 

 

0 G X Y 

 

 

 

 

 

Ž Ž 

op 

since direct limits in mod ., Ab. 

are exact. Using the injectivity of 

Y there exists a morphism Y Y 

making the above 

diagram commutative. Thus the assertion follows. 

 

i 

 

i 

COROLLARY 5. Let 0 X Yi Zi 0, i I, be a directed family of 

exact sequences in modŽ . 

and suppose that i 

is irreducible for all i I. 

Suppose also that : X Y is a morphism in ModŽ ., where Y is 

pure-injectie. Then the direct limit lim i: X Y has the following 

properties: 

Ž. 1 factors through if and only if factors through i 

for all i I. 

Ž. 2 Either factors through or factors through . 

 

Proof. Combine Proposition 4 and Lemma 2. 

QUASI-SERIAL MODULES 

Families of irreducible epimorphisms having isomorphic kernels arise 

quite often for artin algebras of infinite representation type. In fact one 

may ask whether any artin algebra of infinite representation type admits 

such a family. We devote the second half of this paper to a class of finitely 

generated modules where such families naturally occur. To this end recall 

the following concept which was introduced in 8 . A component C of the 

AuslanderReiten quiver of is called quasi-serial if 

Ž. i C contains neither projective nor injective modules; 

Ž . 

k 

ii if 0 X i1Yi 

Z 0 is an almost split sequence of 

modules in C, then k 2 and k 2 with lY Ž . lY Ž . 

1 2 implies that 

lY Ž . lŽ X. lY Ž .. 

1 2 

Examples of such components are the regular components over a hereditary 

artin algebra. An indecomposable module X belonging to a quasiserial 

component is called quasi-serial, and X is quasi-simple if there exists 

no irreducible monomorphism ending in X. We shall use the following 

properties of a quasi-serial module.

418 


Ž X . 

LEMMA 6. 

i i 

Ž. 

To each quasi-serial module X corresponds a unique family 

of indecomposables such that 

1 X1 is quasi-simple and X Xs 

for some s ; 

Ž. 2 there exists an irreducible monomorphism i: Xi Xi1 

for all 

i. 

Proof. See 8 . 

The number s with X Xs 

is called the quasi-length of X and it is 

convenient to put X0 

0. 

LEMMA 7. Let X be a quasi-serial module of quasi-length s and let 

s1 i 

0 i s. Then there is an exact sequence 0 Xi 

Xs 

Ž Tr D. i Ž X . 

si 0. If 1 i s, then the induced morphism 

: X Tr DŽ X . is irreducible. 

s s s1 

Proof. Straightforward. 

Each quasi-simple module determines uniquely a large indecomposable 

module in the following sense. 

THEOREM 8. 

 

Let X be a quasi-simple module. Then there exists a chain 

 

1 2 

XX1 X2 

 

 

of irreducible monomorphisms such that each Xi 

is indecomposable and X 

lim Xi 

is a large denumerably generated indecomposable module. The 

module X 

depends, up to isomorphism, only on X. If Y is a second 

quasi-simple module, then Y X 

if and only if Y X. 

 

 

 

 

 

1 2 

Proof. The existence of a chain X X1 X2 of irreducible 

monomorphisms follows from Lemma 6. For each i 1 define i 

i 

i 

. We obtain a family 0 X X Tr DŽ X . 

i1 1 i i1 0 of ex- 

act sequences where i 

is irreducible for all i by Lemma 7. Thus the 

existence of a large denumerably generated indecomposable module X 

lim Xi 

follows from Theorem 1 and it remains to show its uniqueness. 

First one observes that the X i’s are uniquely determined by X and that 

 

any morphism : X X satisfying i i i1 i1 i i 

is irreducible. The 

latter guarantees that the module lim X constructed in Theorem 1 is 

actually a limit where each morphism Xi Xi1 

is irreducible. Now 

assume an arbitrary choice of irreducible morphisms : X X .It 

 

 

i 

i i i1 

follows from 8, 4.1 

that there exist isomorphisms i: Xi Xi 

satisfying 

 

i1ii i for all i. Taking direct limits the morphism lim i 

gives an isomorphism between lim X i taken over the i’s and lim Xi 

taken over the i ’s. Finally suppose that Y is a second quasi-simple such 

that Y X . The morphism lim : X X composed with the 

i 

isomorphism X Y factors through some Yj 

since X is finitely generated. 

We denote the corresponding morphism by : X Y j. 

It is not hard


to check that there exists an isomorphism : Xj Yj such that j 

and therefore X Y. 

Let C be the center of the artin algebra and denote by I an injective 

envelope of CradŽ C .. Using the functor D Hom Ž , I . 

C between 

Ž . Ž 

op 

Mod and Mod . which induces a duality between modŽ . 

and 

Ž 

op 

mod . we obtain the following consequence of Theorem 8. 

COROLLARY 9. 

Let X be a quasi-simple module. Then there exists a chain 

2 

2 1 

1 

X X X 

 

of irreducible epimorphisms such that each X i is indecomposable and X 

i 

lim X is a large module which depends, up to isomorphism, only on X. 

 

op 

Proof. The -module Y DŽ X. 

is quasi-simple and admits a chain 

 

 

1 2 

YY1Y2 of irreducible monomorphisms. Applying D we obtain 

DŽ 2. DŽ 1. 

a chain DY Ž . DY Ž . 

2 1 X of irreducible epimorphisms 

 

with X lim DY Ž . DŽ lim Y . DY Ž .. 

 

 

 

 

 

i i 

Remark 10. In 8 , it is shown that the module X 

is -pure-injective 

Ž . n Ž . 

 

provided that Tr D X X for some n . Of course, the module X 

Ž . 

op 

is pure-injective since it is of the form DY for some -module Y. We 

do not know of any example where the module X decomposes. 

Let X be a quasi-serial module of quasi-length s and fix a chain X 

 

 

s s1 

Xs Xs1 of irreducible monomorphisms such that Xi 

is inde- 

composable for all i s. We denote by X: X lim Xi X 

the induced 

monomorphism. The module X 

depends only on X according to Lemma 

6 and Theorem 8. Moreover, the morphism X 

is unique in the sense that 

for any different choice of irreducible monomorphisms 

i: Xi X i1, i s, there exists an isomorphism : X X 

with 

. Finally note that induces an exact sequence 

X X X 

 

 

X 

s 

0 X X Ž Tr D. Ž X. 0. 

 

 

 

This follows from Lemma 7. Analogously, there is an induced epimorphism 

X i 

: X X for X lim X which is part of an exact sequence 

 

 

s 

 

X 

 

 

0 Ž D Tr. Ž X. 

X X 0. 

We proceed with a discussion of the morphisms X 

and X . For each 

n and each pair of modules X, Y in modŽ . we denote by rad n 

Ž X, Y . 

the morphisms in Hom Ž X, Y . 

which belong to the nth power of the 

Jacobson radical of mod Ž . . As usual rad 

Ž X, Y . rad n 

Ž X, Y . 

n 

. 

Using the above notation we have the following lemma.

420 


LEMMA 11. Let X Xs 

be a quasi-serial module and Y be indecompos- 

able. If rad n 

Ž X, Y . and n s, then there exists a morphism 

ns 

rad Ž X , Y . such that . 

s1 

 

s 

Proof. First suppose that n s. For each i, 0is, there is an exact 

 

s1 i i 

Ž . 

i 

sequence 0 X XZ 0 with Z Tr D Ž X . 

i i i si accord- 

ing to Lemma 7. Let i be maximal such that there are morphisms 

: Xs1 Y and : Zi Y with s i. The assertion 

follows if i s since 0. Therefore assume that i s. Using the 

s 

Ž . 

i1 

orem 13.3 and Hom X , Z rad Ž X , Z . 

s1 i s1 i . This contradiction 

finishes the first part of our proof. Now suppose that n s. There are 

morphisms : X Y and radŽ Y , Y . 

i i i i i1 for each i such that 

ImŽHom Ž , .. rad i 

Ž X, . and for all i 

i i1 i i 

4, V, 

Lemma 7.10 . Using the first part we find a morphism 0: Xs1 Ys 

such 

that s 0 s. Thus n n 1 s 0 s 

with 

ns 

rad Ž X , Y . and the proof is complete. 

n1 s 0 s1 n 

We are now in position to prove the following property of the morphism 

X : X X . 

THEOREM 12. Let : X Y be a morphism between finitely generated 

indecomposable modules and suppose that X is quasi-serial. 

Ž. 1 rad 

Ž X, Y . if and only if there exists a morphism : X 

Y 

such that X . 

Ž. 2 If rad 

Ž X, Y ., then there exists a morphism : X Y 

such 

that Y X. 

Proof. Ž. 1 The inclusion ImŽHom Ž , Y .. rad 

Ž X, Y . 

X 

is obvious 

from the definition of . Therefore let rad 

Ž X, Y .. We fix a chain 

 

 

1 2 

X 

X1X2 of irreducible monomorphisms with X1 

quasi-simple and 

XXs 

for some s . It follows from the previous lemma that for all 

i any morphism rad 

Ž X , Y . 

i 

i can be extended to a morphism 

rad 

Ž X , Y . 

i1 i1 such that i i1 i. Here one uses the fact that 

rad 

Ž X , Y . rad n 

Ž X , Y . 

i1 i1 for some n . Taking s 

we obtain 

a morphism lim i 

which has the desired property. 

Ž. 2 Similar to Ž. 1 . 

Using again the functor D Hom Ž , I . 

C we obtain the analogous 

property of the morphism X : X X. 

 

COROLLARY 13. Let : Y X be a morphism between finitely generated 

indecomposable modules and suppose that X is quasi-serial. 

property of the almost split sequence starting in Zi 

we deduce that is an 

i1 

Ž . 

i1 

isomorphism. Thus rad X, Y since rad Ž X, Z . by 7, Thei 

i


Ž. 1 rad 

Ž Y, X . if and only if there exists a morphism : Y X 

such that X . 

Ž. 

Ž . 

 

2 If rad Y, X , then there exists a morphism : Y X such 

that Y X . 

REFERENCES 

1. M. Auslander, Representation theory of artin algebras II, Comm. Algebra 1 Ž 1974 ., 

269310. 

2. M. Auslander, Large modules over artin algebras, in ‘‘Algebra, Topology and Categories,’’ 

Academic Press, New YorkLondon, 1976. 

3. M. Auslander and I. Reiten, Representation theory of artin algebras IV, Comm. Algebra 5 

Ž 1977 ., 443518. 

4. M. Auslander, I. Reiten, and S. Smalø, ‘‘Representation Theory of Artin Algebras,’’ 

Cambridge Univ. Press, Cambridge, 1995. 

5. W. Crawley-Boevey, Modules of finite endolength over their endomorphism ring, in 

‘‘Representations of Algebras and Related Topics,’’ Ž S. Brenner and H. Tachikawa, Eds. . 

London Math. Soc. Lect. Note Series, Vol. 168 pp. 127184, Cambridge Univ. Press, 

Cambridge, 1992. 

6. L. Gruson, Simple coherent functors, in ‘‘Representations of Algebras,’’ Lecture Notes in 

Math. Vol. 488, pp. 156159, Springer-Verlag, New YorkBerlin, 1975. 

7. K. Igusa and G. Todorov, A characterization of finite AuslanderReiten quivers, J. 

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8. H. Krause, Generic modules over Artin algebras, Proc. London Math. Soc., to appear. 

9. C. M. Ringel, Finite dimensional hereditary algebras of wild representation type, Math. Z. 

161 Ž 1978 ., 235255.

Constructing Large Modules over Artin Algebras

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