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Revista de la Unión Matemática Argentina

versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.48 n.2 Bahía Blanca jul./dic. 2007

 

The Ladder Construction of Prüfer Modules

Claus Michael Ringel

Dedicated to María Inés Platzeck on the occasion of her 60th birthday

Abstract Let R be a ring (associative, with 1). A non-zero module M is said to be a Prüfer module provided there exists a surjective, locally nilpotent endomorphism with kernel of finite length. The aim of this note is to construct Prüfer modules starting from a pair of module homomorphisms w,v:U0→U1 , where w is injective and its cokernel is of finite length. For R= ℤ the ring of integers, one can construct in this way the ordinary Prüfer groups considered in abelian group theory. Our interest lies in the case that R is an artin algebra.

1. The construction.

Let R be a ring (associative, with 1). The modules to be considered will usually be left R -modules. Our main interest will be the case where R is an artin algebra, however the basic construction should be of interest for any ring R . In fact, the standard examples of what we call Prüfer modules are the Prüfer groups in abelian group theory, thus ℤ -modules. Here is the definition of a Prüfer module: it is a non-zero module P which has a surjective, locally nilpotent endomorphism φ with kernel of finite length. If H is the kernel of φ , we often will write P = H [∞ ] , and we will denote the kernel of φt by H [t] . Observe the slight ambiguity: given a Prüfer module P , not only φ but also all non-trivial powers of φ and maybe many other endomorphisms will have the required properties (surjectivity, local nilpotency, finite length kernel).

The content of the paper is as follows. In the first section we show that any pair of module homomorphisms w, v : U0 → U1 , where w is injective with non-zero cokernel of finite length, gives rise to a Prüfer module. Section 2 provides some examples and section 3 outlines the relationship between Prüfer modules and various sorts of self-extensions of finite length modules. The final sections 4 and 5 deal with degenerations in the sense of Riedtmann-Zwara: we will show that this degeneration theory is intimately connected to the existence of Prüfer modules with some splitting property, and we will exhibit an extension of a recent result by Bautista and Pérez. Our interest in the questions considered here was stimulated by a series of lectures by Sverre Smalø [S] at the Mar del Plata conference, March 2006, and we are indebted to him as well as to M.C.R.Butler and G.Zwara for helpful comments.

For the relevance of Prüfer modules when dealing with artin algebras of infinite representation type, we refer to a forthcoming paper [R5]. The appendix to section 3.3 provides some indications in this direction.

1.1. The basic frame. A pair of exact sequences

0 → U -w→0 U → H → 0 and 0 → K → U -v0→ U → Q → 0 0 1 0 1

yields a module U2 and a pair of exact sequences

0 → U -w→1 U → H → 0 and 0 → K → U -v1→ U → Q → 0 1 2 1 2

by forming the induced exact sequence of 0 → U w-0→ U → H → 0 0 1 using the map v0 :

 0 0 | | |↓ |↓ K ------ K || || ↓ ↓ w0 0 ---→ U0 -- -→ U1 ---→ H ---→ 0 || || ∥∥ ↓v0 ↓v1 ∥ w1 0 ---→ U1 -- -→ U2 ---→ H ---→ 0 || || ↓ ↓ ------ Q| ------ Q| | | ↓ ↓ 0 0

Recall that a commutative square

 f X -- -→ Y1 || || g↓ ↓g ′ Y2 --f-′→ Z

is said to be exact provided it is both a pushout and a pullback, thus if and only if the sequence

 [f] g [g′ -f′] 0 → X --→ Y1 ⊕ Y2 -----→ Z → 0

is exact. Note that our basic setting provides an exact square

 U --w0-→ U |0 |1 v0|↓ |↓v1 U1 -- -→ U2 w1

Next, we will use that the composition of exact squares is exact:

(E1) The composition of two exact squares

X ---→ Y1 - --→ Z1 || || || ↓ ↓ ↓ Y2 ---→ Z2 - --→ A

yields an exact square

X -- -→ Z | | 1 |↓ |↓ Y2 -- -→ A

1.2. The ladder. Using induction, we obtain in this way modules Ui and pairs of exact sequences

 wi vi 0 → Ui-→ Ui+1 → H → 0 and 0 → K → Ui -→ Ui+1 → Q → 0

for all i ≥ 0.

We may combine the pushout diagrams constructed inductively and obtain the following ladder of commutative squares:

 w0 w1 w2 w3 U0 - --→ U1 ---→ U2 -- -→ U3 ---→ ⋅⋅⋅ || || || || v0↓ v1↓ v2↓ v3↓ w1 w2 w3 w4 U1 - --→ U2 ---→ U3 -- -→ U4 ---→ ⋅⋅⋅

We form the inductive limit  ⋃ U∞ = iUi (along the maps wi ).

Since all the squares commute, the maps vi induce a map U∞ → U∞ which we denote by v ∞ :

 w0 w1 w2 w3 ⋃ U|0 ---→ U1| ---→ U|2 -- -→ U3| ---→ ⋅⋅ ⋅ iU|i = U ∞ v0| v1| v2| v3| | v∞ ↓ ↓ ↓ ↓ ↓ w1 w2 w3 w4 ⋃ U1 ---→ U2 ---→ U3 -- -→ U4 ---→ ⋅⋅ ⋅ iUi = U ∞

We also may consider the factor modules U∞ ∕U0 and U ∞∕U1 . The map v∞ : U∞ → U ∞ maps U0 into U1 , thus it induces a map

v-: U ∞∕U0 -→ U∞ ∕U1.

Claim. The map -- v is an isomorphism. Namely, the commutative diagrams

 wi-1 0 --- → Ui-1 -- -→ Ui - --→ H - --→ 0 || || ∥∥ ↓vi-1 ↓vi ∥ wi 0 --- → Ui -- -→ Ui+1 - --→ H - --→ 0

can be rewritten as

 wi-1 0 -- -→ Ui- 1 ---→ Ui ---→ Ui∕Ui-1 -- -→ 0 || || || ↓vi-1 ↓vi ↓vi wi 0 -- -→ Ui ---→ Ui+1 ---→ Ui+1∕Ui -- -→ 0

with an isomorphism v- : U ∕U → U ∕U . i i i- 1 i+1 i The map v- is a map from a filtered module with factors Ui∕Ui- 1 (where i ≥ 1 ) to a filtered module with factors Ui+1∕Ui (again with i ≥ 1 ), and the maps -- vi are just those induced on the factors.

It follows: The composition of maps

 p v-1 U ∞∕U0 ---→ U∞ ∕U1 ---→ U ∞ ∕U0

with p the projection map is an epimorphism φ with kernel U1∕U0. It is easy to see that φ is locally nilpotent, namely we have φt(Ut∕U0 ) = 0 for all t .

Summary. (a) The maps vi yield a map

v∞ : U∞ → U ∞

with kernel K and cokernel Q .

(b) This map v∞ induces an isomorphism -- v : U ∞ ∕U0 → U∞ ∕U1 . Composing the inverse of this isomorphism with the canonical projection p , we obtain an endomorphism  ---1 φ = (v) ∘ p

 p v- 1 U ∞ ∕U0 -→ U∞ ∕U1 --→ U∞ ∕U0.

If the cokernel H of w 0 is non-zero and of finite length, then U ∕U ∞ 0 is a Prüfer module with respect to φ, with basis H ; in this case, we call U ∞ ∕U0 (or better the pair (U ∞∕U0, φ) ) the Prüfer module defined by the pair (w0, v0) or by the ladder Ui . Prüfer modules which are obtained in this way will be said to be of ladder type.

If necessary, we will use the following notation: Ui (w0, v0) = Ui , for all i ∈ ℕ ∪ {∞ } and P (w0, v0) = U ∞ ∕U0 for the Prüfer module. Since P (w0,v0) is a Prüfer module with basis the cokernel H of w0 , we will sometimes write H [n ] = Un ∕U0 or even H [n; w0,v0].

Remark: Using a terminology introduced for string algebras [R3], we also could say: U ∞ is expanding, U∞ ∕U0 is contracting.

Lemma. Assume that P = P (w, v) with w, v : U0 → U1 . Then P is generated by U1 .

Proof: For i ≥ 2 , the module U i is a factor module of U ⊕ U , i-1 i-1 thus by induction, Ui is generated by U1 .

1.3. The chessboard. Assume now that both maps w0,v0 : U0 → U1 are monomorphisms. Then we get the following arrangement of commutative squares:

 w0 w1 w2 w3 U0 ---→ U1 ---→ U2 ---→ U3 ---→ ⋅⋅⋅ || || || || v0↓ v1↓ v2↓ v3↓ w1 w2 w3 U1 ---→ U2 ---→ U3 ---→ ⋅⋅⋅ || || || v1↓ v2↓ v3↓ w2 w3 U2 ---→ U3 ---→ ⋅⋅⋅ || || v2↓ v3↓ w3 U3 ---→ ⋅⋅⋅ || v3↓ ⋅⋅⋅

Note that there are both horizontally as well as vertically ladders: the horizontal ladders yield U∞ (w0,v0) (and its endomorphism v∞ ); the vertical ladders yield U ∞ (v0,w0) (and its endomorphism w∞ ).

2. Examples.

(1) The classical example: Let R = ℤ be the ring of integers, and U0 = U1 = ℤ its regular representation. Module homomorphisms ℤ → ℤ are given by the multiplication with some integer n , thus we denote such a map just by n . Let w0 = 2 and v0 = n . If n is odd, then P (2,n) is the ordinary Prüfer group for the prime 2 , and U ∞(2,n ) = ℤ[1] 2 (the subring of ℚ generated by 1 2 ). If n is even, then P (2,n) is an elementary abelian 2-group.

(2) Let R = K (2) be the Kronecker algebra over some field k . Let U0 be simple projective, U 1 indecomposable projective of length 3 and w0 : U0 → U1 a non-zero map with cokernel H (one of the indecomposable modules of length 2). The module P (w0,v0) is the Prüfer module for H if and only if v0 ∕∈ kw0, otherwise it is a direct sum of copies of H .

(3) Trivial cases: First, let w be a split monomorphism. Then the Prüfer module with respect to any map v : U0 → U1 is just the countable sum of copies of H . Second, let w : U0 → U1 be an arbitrary monomorphism, let β : U1 → U1 be an endomorphism. Then P (w, βw ) is the countable sum of copies of H .

(4) Assume that there exists a split monomorphism v : U0 → U1 , say U1 = U0 ⊕ X and  1 v = [0] : U0 → U1 . Then

 w 0 → U0 -→ U0 ⊕ X → H → 0

is a Riedtmann-Zwara sequence as discussed in section 4, thus H is a degeneration of X .

Remark: Not all Prüfer modules are of ladder type. Consider the generalized Kronecker algebra Λ with countably many arrows α0, α1,... starting at the vertex a and ending in the vertex b . Define a representation P = (Pa, Pb,αi)i as follows: Let Pa = Pb be a vector space with a countable basis e0,e1,... and let αi : Pa → Pb be defined by αi(ej) = ej-i provided j ≥ i and αi (ej) = 0 otherwise. Let φa,φb be the endomorphism of Pa of Pb , respectively, which sends e0 to 0 and e i to e i- 1 for i ≥ 1. Then P is a Prüfer module (with respect to φ , but also with respect to any power of φ ). Obviously, P is a faithful Λ -module. Assume that P = P (w, v) for some maps w,v : U0 → U1 with U0, U1 of finite length. Then P is generated by U1 , according to Lemma 1.2. However U1 is of finite length and no finite length Λ -module is faithful.

3. Ladder extensions.

3.1. The definition. Let H be a non-zero module of finite length. A self-extension 0 → H → H [2] → H → 0 is said to be a ladder extension provided there is a commutative diagram with exact rows

 q 0 ---→ U0 -- -→ U1 ---→ H ---→ 0 || || ∥∥ ↓f ↓ ∥ 0 ---→ H -- -→ H [2] ---→ H ---→ 0

such that f factors through q , say f = qv for some v : U0 → U1.

This means that we have a commutative diagram with exact rows of the following kind (here f = qv0 ):

 w0 q 0 ---→ U0 ---→ U1 -- -→ H -- -→ 0 || || ∥∥ v0↓ v1↓ ∥ w1 0 ---→ U1 ---→ U2 -- -→ H -- -→ 0 || || ∥∥ q↓ ↓ ∥ 0 ---→ H ---→ H [2 ] -- -→ H -- -→ 0.

Thus, in order to construct all the ladder extensions of H , we may start with an arbitrary epimorphism q : U1 → H, form its kernel w0 and consider any homomorphism v : U → U 0 0 1 .

According to section 1 we know: Ladder extensions build up to form Prüfer modules.

Lemma. Let k be a commutative ring and Λ a k -algebra. Then H [2;w0, v0] = H [2;w0,v0 + μw0 ] for any μ ∈ k.

Proof: We deal with the exact sequence induced by qv0 or q(v0 + μw0 ) , respectively. But q(v0 + μw0 ) = qv0 + qμw0 = qv0, since qw0 = 0.

Also, any central automorphism λ of U0 yields isomorphic extensions H [2;w0,v0] and H [2;w0, λv0] . This shows that the extension H [2;w0,v0] only depends on the k -subspace ⟨w0, v0⟩.

Remark. Not all self-extensions are ladder extensions. For example: A non-zero self-extension of a simple module S over an artinian ring is never a ladder extension!

Proof: Construct the corresponding ladder, thus the corresponding Prüfer module S [∞ ] . The module S [n] would be a (serial) module of Loewy length n , with n arbitrary. But the Loewy length of any module over the artinian ring R is bounded by the Loewy length of RR, thus S[∞ ] cannot exist.

Example. Here is a further example of a self-extension which is not a ladder extension. Consider the following quiver Q

 .............................................................................. a∘................................b...................................∘ ....... β............................... ............................................................................................................................

with one loop β at the vertex b, and one arrow from a to b . We consider the representations of Q with the relation β3 = 0. The universal covering ^ Q of Q has many D5 subquivers  ′ Q of the form

 ∘............. ∘............. ...................................................... ...................................................... ∘ ...................................................................∘...................................................................∘

and we consider some representations of Q′ ; we present here the corresponding dimension vectors.

 1........ 0........ 1........ 1........ 0........ 1........ ......................................... ......................................... .......................................... .......................................... .......................................... .......................................... 1...................................................................1...................................................................................0.................... 1 ...................................................................2....................................................................................1................. 0........................................................1.................. 1................. ′ ′′ H H H

There is an obvious exact sequence

 ′ ′′ 0 → H → H → H → 0.

Under the covering functor, the representations H and  ′′ H are identified, thus we obtain a self-extension. One easily checks that this self-extension is not a ladder extension.

Proposition. Let H be an indecomposable module with Auslander-Reiten translate isomorphic to H . Assume that there is a simple submodule S of H with  1 Ext (S, S) = 0. Then the Auslander-Reiten sequence ending (and starting) in H is a ladder extension.

Proof. Let  ′ 0 → H → H → H → 0 be the Auslander-Reiten sequence. Denote by u : S → H the inclusion map. Since the map H → H ∕S factors through  ′ H → H , there is a commutative diagram with exact rows of the following form:

 w q 0 ---→ S ---→ U ---→ H ---→ 0 || || ∥∥ u↓ ↓ ∥ ′ 0 ---→ H ---→ H ---→ H ---→ 0

Now form the induced exact sequence:

0 - --→ S ---→ U ′ -- -→ S --- → 0 ∥ | | ∥∥ |↓ u|↓ w q 0 - --→ S ---→ U -- -→ H --- → 0

Since Ext1 (S,S ) = 0, the induced sequence splits, thus we obtain a map v : S → U with qv = u. It follows that  ′ H = H [2;w,v ].

We do not know whether one can delete the assumption about the existence of S .

3.2. Standard self-extensions.

Let H be an R -module, say with an exact sequence 0 → ΩH -→u P H p-→ H → 0 , where P H denotes a projective cover of H . We know that

 1 Ext (H, H ) = Hom--(ΩH, H ) = Hom (ΩH, H )∕ Im (Hom (u,H )).

Note that

Im (Hom (u,H )) ⊆ Im(Hom (ΩH, p)) ⊆ Hom (ΩH, H ).

(Proof: Hom (u,H ) : Hom (P H, H ) → Hom (ΩH, H ) , thus take φ : P H → H and form φu. Since p : P H → H is surjective and P H is projective, there is φ′ : P H → P H with φ = pφ′. Thus φu = pφ′u is in the image of Hom (ΩH, p). )

Thus we can consider

 1 Ext (H, H )s := Im (Hom (ΩH, p))∕Im (Hom (u, H ))

as a subgroup of Hom (ΩH, H )∕Im (Hom (u, H )) = Ext1 (H, H ). We call the elements of Ext1(H, H )s the standard self-extensions.

Proposition. Standard self-extensions are ladder extensions.

Proof. Here is the usual diagram in which way a map f : ΩH → H yields a self-extension of H

 u p 0 -- -→ ΩH| ---→ P|H -- -→ H∥ -- -→ 0 f|↓ |↓ ∥∥ 0 -- -→ H ---→ H [2 ] -- -→ H -- -→ 0

The standard extensions are those where the map f factors through p , say  ′ f = pw with  ′ w : ΩH → P H :

 u p 0 -- -→ ΩH| ---→ P|H -- -→ H∥ -- -→ 0 ′| w ′| ∥ w↓ 1↓ ∥ u1 0 -- -→ P|H ---→ U|2 -- -→ H∥ -- -→ 0 p| | ∥ ↓ ↓ ∥ 0 -- -→ H ---→ H [2 ] -- -→ H -- -→ 0

3.3. Modules of projective dimension 1.

Proposition. If the projective dimension of H is at most 1, then any self-extension of H is standard, thus a ladder extension.

Proof: Consider a module H with a projective presentation  p 0 → P ′ → P -→ H → 0. Any self-extension of H is given by a diagram of the following kind:

 u p 0 ---→ P ′-- -→ P ---→ H ---→ 0 | | ∥ f|↓ |↓ ∥∥ 0 ---→ H -- -→ H [2] ---→ H ---→ 0

Since P ′ is projective and p : P → H surjective, there is a map f ′ : P ′ → P such that f = pf ′. The self-extension is given just by  ′ H [2] = H [2;u, f ].

Corollary. If R is a hereditary ring, any self-extension is standard, thus a ladder extension.

Example of a ladder extension which is not standard. Consider the quiver Q

 ∘a ......................... ......................... α....................................................................................β ..............∘b.............. γ..........................................................................................................δ ∘c

such that δα = 0 = γβ = γ α - δβ. Consider the indecomposable length 2 module H = (β : a → b) annihilated by α . Then the kernel ΩH of PH → H is ΩH = (γ : b → c). We may visualize this as follows:

 a a b u ...α..................∙........................................................β....... p ∙..........................................β ............................................................ ∙.................................. ................................................................................................................. ∙........................................... ..................∙......................... ................................................................................................................. ∙......... γ.∙................c γ ∙..............c....................δ

There is a ladder extension of H , given by the non-trivial map f : ΩH → H , but this map does not factor through P H , since Hom (ΩH, PH ) is one-dimensional, generated by u . Note that -- f : ΩH ∕K → H factors through p : PH ∕u (K ) → H , where K = S (c) is the kernel of f .

Appendix. Here, we want to indicate that the Corollary can be used in order to obtain a conceptual proof of the second Brauer-Thrall conjecture for hereditary artin algebras.

Assume that there is no generic module. We show: Any indecomposable module is a brick without self-extensions. Assume that there is an indecomposable module M which is not a brick or which does have self-extensions. If M is not a brick, then the brick paper [R2] shows that there are bricks M ′ with self-extensions. Thus, we see that there always is a brick H with self-extensions. Take any non-zero self-extension of H . According to 3.2, such a self-extension is standard, thus a ladder extension, thus we obtain a corresponding Prüfer module H [∞ ] . The process of simplification [R1] shows that all the modules H [n ] are indecomposable. Thus H [∞ ] is not of finite type and therefore there exists a generic module [R5].

But if any indecomposable module is a brick without self-extensions, the quadratic form is weakly positive. Ovsienko asserts that then there are only finitely many positive roots, thus the algebra is of bounded representation type and therefore of finite representation type.

3.4. Warning. A Prüfer module H [∞ ] is not necessarily determined by H [2] , even if it is of ladder type.

As an example take the generalized Kronecker quiver with vertices a,b and three arrows α, β,γ : a → b . and let H be the two-dimensional indecomposable representation annihilated by α and β . Consider a projective cover q : P H → H , let ΩH be its kernel, say with inclusion map w : ΩH → P H.

0 → ΩH -w→ PH  q -→ H → 0 (*)

The ladders to be considered are given by the various maps f : ΩH → P H such that the image of f is not contained in ΩH (otherwise, the induced self-extension of H will split). In order to specify a self-extension H [2] of H , we require that H [2 ] is annihilated say by γ .

We will consider several copies of PH . If ei ∈ (PH )a is a generator, let us denote e = α(e ), e = β(e ), e = γ(e ), i1 i i2 i i3 i thus, e ,e ,e i1 i2 i3 is a basis of (PH )b .

We start with P H generated by e1 and consider the exact sequence (* ) as displayed above. We see that e12,e13 is a basis of ΩH.

Now, let us consider two maps f,g : ΩH → PH , here we denote the generator of PH by e . 0 The first map f is given by f (e ) = e 12 01 and f (e13) = 0. The second map g : ΩH → P H is defined by g(e12) = e01 and g(e13) = e02 .

Note that qf = qg , thus H [2;w, f] = H [2;w, g] and actually this is precisely the self-extension of H annihilated by γ.

An easy calculation shows that H [3;w, f] (and even H [∞; w,f ] ) is annihilated by γ , whereas H [3;w, g] is faithful. The following displays may be helpful; always, we exhibit the modules:

 w0 w1 U0 =|ΩH --- → U1 =|P H -- -→ U2| |v |v |v ↓ 0 ↓ 1 ↓ 2 w1 w2 U1 =|P H --- → U2| -- -→ U3| |q | | ↓ ↓ ↓ H --- → H [2] -- -→ H [3]

First the display for the homomorphism f .

 e.......1.... ..e...........1......................e2...................... .............................................................................................................. ......................................................................................... .............................................................................................................. e........................e.........................e.....................................e.................... e13 e12 e13 e12 e11 13 12 e1212 21 ................ ................ ................ ................................................... ................................................... ................................................... .............e0................................. ...... ...............e...........0.......................................e.....1............................................ ...... ..........................e0............................................e..................1..........................e2............................................ e03............... e.....0..............2............e0........1 ........................................................................................................ e03...........e02........ e...........01........ e.........1..1 ........................................................................................................ e03...........e0........2 e.......0............1..e1.........1........ e...........21 e12 e12 e22 ......................... ......................... ......................... .......................................... .......................................... .......................................... e0 e1 e0 e1 e2 e0.............................. .............................................................................................................. .................................................................................................... .............................. ................................................................................................................................................................... e.......01 ee0112 e11 ee0112 ee1212 e21

Now the corresponding display for the homomorphism g .

 e1 e2 .............................................................................................................. .............................e...........................1................................. ................................... ........................................................................................................................................................................................................ e13 e12 e13.... e1...2 e1....1 e13 ee1223 ee1212 e21 ....... ....... ....... ....................................... ....................................... ....................................... ..................... ..................... ..................... e0........... ..e...........0......................e.............1...................... ......................e0........................................................e....................1...................................e.........2...................................... ......................................................................................... ........................................................................... e........................e..............................................e........................................e.................... e03...............e0..........2.............. e............0..............1.....................e1.........1...........e..............21 e03 e02 e01 03 e0213 e0112 11 e13 ee1223 e22 ................ ................ ................ ................................................... ................................................... ................................................... e e e e0................. ....... e0..........................e..................1.......................... ....... 0...................................1...............................................................................................2............................................................ ....e................01 ....................................................................................................... e......0.............1.e1..........1 ....................................................................................................... ee..0112... ee..12.12 e2..1 e12 e23

4. Degenerations.

Definition: Let X, Y be finite length modules. Call Y a degeneration of X provided there is an exact sequence of the form 0 → U → X ⊕ U → Y → 0 with U of finite length. (such a sequence will be called a Riedtmann-Zwara sequence). The map U → U is called a corresponding steering map. (Note that in case we deal with modules over a finite dimensional k -algebra and k is an algebraically closed field, then this notion of degeneration coincides with the usual one, as Zwara [Z2] has shown.)

The proof of the following result is essentially due to Zwara, he used this argument in order to show that Y is a degeneration of X if and only if there is an exact sequence 0 → Y → X ⊕ V → V → 0 (a co-Riedtmann-Zwara sequence) with V of finite length.

Proposition. Let X, Y be Λ -modules of finite length. The following conditions are equivalent:

(1) Y is a degeneration of X .
(2) There is a Prüfer module Y [∞ ] and some natural number t0 such that Y [t + 1] ≃ Y [t] ⊕ X for all t ≥ t0.
(3) There is a Prüfer module Y [∞ ] and some natural number t 0 such that Y [t + 1] ≃ Y [t ] ⊕ X 0 0 .

Here is the recipe how to obtain a Prüfer module Y [∞ ] starting from a degeneration: If Y is a degeneration of X , say with steering module U , then there exists a monomorphism μ : U → U ⊕ X with cokernel Y . The Prüfer module Y [∞ ] we are looking for is

Y [∞ ] = P(μ, [10]).

Proof of the implication (3) =⇒ (1). Assume that there is a Prüfer module Y [∞ ] such that Y[t + 1 ] ≃ Y [t] ⊕ X. We get the following two exact sequences

0 → Y [t] → Y [t + 1] → Y [1] → 0, 0 → Y [1] → Y [t + 1] → Y [t] → 0,
(1)

in the first, the map Y[t + 1 ] → Y [1] is given by applying ψt , in the second the map Y [t + 1] → Y[t] is given by applying ψ. In both sequences, we can replace Y [t + 1] by Y [t] ⊕ X. Thus we obtain as first sequence a new Riedtmann-Zwara sequence, and as second sequence a dual Riedtmann-Zwara sequence:

0 → Y [t] → Y[t] ⊕ X → Y → 0, 0 → Y → Y [t] ⊕ X → Y [t] → 0,
(2)

note that both use the same steering module, namely Y [t]. Thus:

Remark. We see: The module Y is a degeneration of X if and only if there exists a module V and an exact sequence 0 → Y → V ⊕ X → V → 0 .

Proof of the proposition. We need further properties of exact squares:

(E2) For any map a : U → V , and any module X , the following diagram is exact:

 a U -- -→ V || || [10]↓ ↓[10] U ⊕ X --a⊕-1→ V ⊕ X. X

(E3) Let

 f X| ---→ Y1| | | 0↓ ↓ Y ---→ Z 2 f′

be exact. Then f ′ is split mono.
(E4) Assume that we have the following exact square

 a U ---→ V || || b↓ b′↓ W ---a→′ X

and that b is a split monomorphism, then the sequence

 [ab] [b′ a′] 0 → U --→ V ⊕ W ---→ X → 0

splits.

Proofs. (E2) is obvious. (E3): Since [ ] f0 is injective, f : X → Y1 is injective. Let Q be the cokernel of f . We obtain the map f ′ by forming the induced exact sequence of  f 0 → X -→ Y1 → Q → 0 , using the zero map X → Y1 . But such an induced exact sequence splits. (E4) Assume that pb = 1U . Then [0 p][ab] = 1U.

There is the following lemma (again, see Zwara [Z1]):

Lemma (Existence of nilpotent steering maps.) If there is an exact sequence 0 → U → X ⊕ U → Y → 0, then there is an exact sequence 0 → U ′ → X ⊕ U′ → Y → 0 such that the map U′ → U ′ is nilpotent.

Proof: We can decompose  ′ ′ U = U1 ⊕ U2 = U1 ⊕ U 2 such that the given map f : U → U maps U1 into  ′ U1 , U2 into  ′ U 2 and such that the induced map f1 : U1 → U ′1 belongs to the radical of the category, whereas the induced map f2 : U2 → U ′2 is an isomorphism. We obtain the following pair of exact squares

 [10] U1 ---→ U1 ⊕ U2 -- -→ X || || || f1↓ f1⊕f2↓ ↓ ′ ′ ′ U 1 ---1→ U1 ⊕ U2 -- -→ Y [0]

(the left square is exact according to (E2)). The composition of the squares is the desired exact square (note that U ′ 1 is isomorphic to U1 ).

Assume that a monomorphism  [ ] φ w = g : U → U ⊕ X with cokernel Y and  t φ = 0 is given. Consider also the canonical embedding  [ ] v = 1 : U → U ⊕ X 0 and form the ladder Ui (w, v) for this pair of monomorphisms w, v . The modules Y [i] = Ui(w,v )∕U0 (w,v ) are just the modules we are looking for: As we know, there is a Prüfer module (Y [∞ ],ψ ) with Y [i] being the kernel of ψi.

We construct the maps wn, vn explicitly as follows:

 ⌊ ⌋ φ [w ] wn = ⌈ g ⌉ = : U ⊕ Xn → (U ⊕ X ) ⊕ Xn 1Xn 1Xn

and

 [ ] v = 1U⊕Xn : U ⊕ Xn → U ⊕ Xn ⊕ X, n 0

using the recipe (E2). Thus we obtain the following sequence of exact squares:

 [ ] ⌊ φ ⌋ [φ] φg ⌈ g 1 ⌉ U ------g------- U⊕X -----1----- U⊕X ⊕X ----1---U ⊕X ⊕X ⊕X ------------ | | ⌊ ⌋| ⌊ ⌋ | [1]| [11]| ⌈1 1 ⌉| |⌈1 11 |⌉ | 0 | 00 | 0 010 | 0 0010 | ------------ --------- ------ ----------- U ⊕X [φ ] U⊕X ⊕X ⌊ φ ⌋ U ⊕X ⊕X ⊕X ⌊φ ⌋U⊕X ⊕X ⊕X ⊕X g 1 ⌈ g 1 ⌉ |⌈g 1 |⌉ 1 11

In particular, we have  n Un = Un (w,v ) = U ⊕ X .

Note that the composition  n wn -1⋅⋅⋅w0 : U → U ⊕ X is of the form [ ] φn gn for some  n gn : U → X .

We also have the following sequence of exact squares:

 w0 w1 w2 w3 U =| U0 -- -→ U|1 -- -→ U|2 - --→ U3| ---→ ⋅⋅⋅ | | | | ↓ ↓ ↓ ↓ s1 s2 s3 0 -- -→ Y [1 ]-- -→ Y [2]- --→ Y [3] ---→ ⋅⋅⋅

where the vertical maps are of the form

 [hn qn] Un = U ⊕ Xn ----→ Y [n].

The composition of these exact squares yields an exact square

 wn-1⋅⋅⋅w0 n U --- --→ U ⊕ X || || ↓ ↓[hn qn] 0 -- -→ Y [n]

Here we may insert the following observation: This sequence shows that the module Y [n ] is a degeneration of the module Xn .

Since the composition wn- 1⋅⋅⋅w0 : U → U ⊕ Xn is of the form [ ] φn g , n and φt = 0, it follows that ht is a split monomorphism, see (E3).

Also, we can consider the following two exact squares, with  [ ] φ w = g : U → V = U ⊕ X (the upper square is exact, according to (E2)):

 U -- w-→ V | | [10]|↓ |↓ [10 ] [w 1] U ⊕ Xt -- -→ V ⊕ Xt || || [ht qt]↓ ↓ [ht+1 qt+1] Y[t] -- -→ Y [t+1 ]

The vertical composition on the left is ht , thus, as we have shown, a split monomorphism. This shows that the exact sequence corresponding to the composed square splits (E4): This yields

U ⊕ Y [t+1 ] ≃ Y [t] ⊕ V = Y [t] ⊕ U ⊕ X.

Cancelation of U gives the desired isomorphism:

Y[t+1 ] ≃ Y[t] ⊕ X.

Remark to the proof. Given the Riedtmann-Zwara sequence

 [φ] g 0 → U --→ U ⊕ X → Y → 0,

we have considered the following pair of monomorphisms

 ′ [ φ] w = [10],w = g : U → U ⊕ X.

The corresponding Prüfer modules are X (∞ ) and Y [∞ ] , respectively. And  ′ n Un (w,w ) = U ⊕ X . As we know, we can assume that φ is nilpotent. Then all the linear combinations

 ′ [ 1+ λφ] w + λw = g

with λ ∈ k are also split monomorphisms (with retraction [η 0] , where η = (1 + λφ )-1 ).

Corollary. Assume that Y is a degeneration of X . Then there exists a Prüfer module Y[∞ ] such that Y [∞ ] is isomorphic to Y [t] ⊕ X (∞) for some natural number t .

5. Application: The theorem of Bautista-Pérez.

Here we assume that we deal with an artin algebra Λ , and all the modules are Λ -modules of finite length.

Proposition. Let W be a module with  1 Ext (W, W ) = 0 and assume there is given an exact sequence 0 → U → V → W → 0 . Then the cokernel of any monomorphism U → V is a degeneration of W .

Corollary (Bautista-Pérez). Let U, V be modules, and let W and W ′ be cokernels of monomorphisms U → V. Assume that both Ext1(W, W ) = 0 and Ext1(W ′,W ′) = 0. Then the modules W and  ′ W are isomorphic.

Both assertions are well-known in case k is an algebraically closed field: in this case, the conclusion of the proposition just asserts that  ′ W is a degeneration of W in the sense of algebraic geometry. The main point here is to deal with the general case when Λ is an arbitrary artin algebra. The corollary stated above (under the additional assumptions that V is projective and that w(U ),w′(U ) are contained in the radical of V ) is due to Bautista and Pérez [BP] and this result was presented by Smalø with a new proof [S] at Mar del Plata.

We need the following well-known lemma.

Lemma. Let W be a module with  1 Ext (W, W ) = 0. Let U0 ⊂ U1 ⊂ U2 ⊂ ⋅⋅⋅ be a sequence of inclusions of modules with Ui∕Ui -1 = W for all i ≥ 1. Then there is a natural number n0 such that U ⊂ U n n+1 is a split monomorphism for all n ≥ n . 0

Let us use it in order to finish the proof of the proposition. Let U = U,U = V, 0 1 and w : U → V 0 0 0 the given monomorphism with cokernel W . Let v0 : U0 → U1 be an additional monomorphism, say with cokernel  ′ W . Thus we are in the setting of section 1. We apply the Lemma to the chain of inclusions

 w0 w1 w2 U0 -→ U1 -→ U2 -→ ⋅⋅⋅

and see that there is n such that wn : Un → Un+1 splits. This shows that U n+1 is isomorphic to U ⊕ W. n But we also have the exact sequence

 vn 0 → Un -→ Un+1 → W ′ → 0.

Replacing U n+1 by U ⊕ W n , we see that we get an exact sequence of the form

 v 0 → Un -n→ Un ⊕ W → W ′ → 0

(a Riedtmann-Zwara sequence), as asserted.

Proof of the Corollary. It is well-known that the existence of exact sequences

0 → X → X ⊕ W → W ′ → 0 and 0 → Y → Y ⊕ W ′ → W → 0

implies that the modules W and  ′ W are isomorphic [Z1]. But in our case we just have to change one line in the proof of the proposition in order to get the required isomorphism. Thus, assume that both Ext1(W, W ) = 0 and Ext1 (W ′,W ′) = 0 . Choose n such that both the inclusion maps

w : U → U and v : U → U n n n+1 n n n+1

split. Then Un+1 is isomorphic both to Un ⊕ W and to  ′ Un ⊕ W , thus it follows from the Krull-Remak-Schmidt theorem that W and  ′ W are isomorphic.

Remark. Assume that w, w′ : U → V are monomorphisms with cokernels W and W ′ , respectively, and that Ext1 (W, W ) = 0 and Ext1(W ′,W ′) = 0. Then w splits if and only if w′ splits.

Proof: According to the corollary, we can assume W = W ′ . Assume that w splits, thus V is isomorphic to U ⊕ W . Look at the exact sequence  w′ 0 → U -→ V → W → 0 . If it does not split, then dim End (V) < dim End (U ⊕ W ) , but V is isomorphic to U ⊕ W.

As we have mentioned, the lemma is well-known; an equivalent assertion was used for example by Roiter in his proof of the first Brauer-Thrall conjecture, a corresponding proof can be found in [R4]. We include here a slightly different proof:

Applying the functor Hom (W, - ) to the short exact sequence  wi-1 0 → Ui-1 --→ Ui → W → 0, we obtain the exact sequence

Ext1(W, Ui- 1) → Ext1 (W, Ui ) → Ext1 (W, W ).

Since the latter term is zero, we see that we have a sequence of surjective maps

 1 1 1 Ext (W, U0) → Ext (W, U1) → ⋅⋅⋅ → Ext (W, Ui ) → ⋅⋅⋅ ,

being induced by the inclusion maps U0 → U1 → ⋅⋅⋅ → Ui → ⋅⋅⋅ . The maps between the Ext -groups are k -linear. Since Ext1 (W, U0) is a k -module of finite length, the sequence of surjective maps must stabilize: there is some n0 such that the inclusion Un → Un+1 induces an isomorphism

Ext1 (W, U ) → Ext1(W, U ) n n+1

for all n ≥ n0. Now we consider also some Hom -terms: the exactness of

Hom (W, Un+1) → Hom (W, W ) → Ext1 (W, Un ) → Ext1 (W, Un+1 )

shows that the connecting homomorphism is zero, and thus that the map Hom (W, Un+1 ) → Hom (W, W ) (induced by the projection map p : Un+1 → W ) is surjective. But this means that there is a map h ∈ Hom (W, Un+1 ) with ph = 1W , thus p : Un+1 → W is a split epimorphism and therefore the inclusion map Un → Un+1 is a split monomorphism.

Remark. In general, there is no actual bound on the number n0 . However, in case of dealing with the chain of inclusions

 w0 w1 wn U0 -→ U1 -→ U2 -→ ⋅⋅⋅

such a bound exists, namely the length of Ext1 (W, U0 ) as a k -module, or, even better, the length of Ext1(W, U ) 0 as an E -module, where E = End (W ).

Proof: Look at the surjective maps

Ext1 (W, U ) → Ext1(W, U ) → ⋅⋅⋅ → Ext1(W, U ) → ⋅⋅⋅ , 0 1 i

being induced by the maps Un -wn→ Un+1 (and these maps are not only k -linear, but even E -linear). Assume that Ext1(W, U ) → Ext1 (W, U ) n n+1 is bijective, for some n . As we have seen above, this implies that the sequence

0 → Un  w -→n Un+1W → 0 (*)

splits. Now the map wn+1 is obtained from (*) as the induced exact sequence using the map w ′n . With (*) also any induced exact sequence will split. Thus w n+1 is a split monomorphism (and  1 1 Ext (W, Un+1 ) → Ext (W, Un+2) will be bijective, again). Thus, as soon as we get a bijection  1 1 Ext (W, Un ) → Ext (W, Un+1 ) for some n , then also all the following maps  1 1 Ext (W, Um ) → Ext (W, Um+1 ) with m > n are bijective.

Example. Consider the D4 -quiver with subspace orientation:

 .............................................................b a.............................................................................................................c d......

and let Λ be its path algebra over some field k . We denote the indecomposable Λ -modules by the corresponding dimension vectors. Let

 10 0 2 11 1 11 ′ 1 01 0 10 U0 = 0, U1 = 1, W = 1, W = 1 ⊕ 0.

Note that a map w0 : U0 → U1 with cokernel W exists only in case the base-field k has at least 3 elements; of course, there is always a map  ′ w 0 : U0 → U1 with cokernel  ′ W .

We have dim Ext1(W, U0 ) = 2, and it turns out that the module U2 is the following:

U2 = 10 1 ⊕ 1 10 ⊕ 1 11. 1 1 0

The pushout diagram involving the modules U0, U1 (twice) and U2 is constructed as follows: denote by μa,μb, μc monomorphisms U0 → U1 which factor through the indecomposable projective modules P (a),P (b),P (c) , respectively. We can assume that μc = - μa - μb , so that a mesh relation is satisfied. Denote the 3 summands of U 2 by M ,M ,M a b c , with non-zero maps νa : U1 → Ma, νb : U1 → Mb, νc : U1 → Mc, such that νaμa = 0, νbμb = 0, νcμc = 0. There is the following commutative square, for any q ∈ k, we are interested when q ∕∈ {0,1} :

 w =μ +qμ U0 ----0--a---b---U1 | | [ ] v0=μa | v1= ν0b | | νc U1 -----[------]--U2 w1= ννab (1- q)νc

(the only calculation which has to be done concerns the third entries: ν (μ + qμ ) = (1 - q)ν μ c a b c a ). Note that w 1 (as well as w′ 1 ) does not split.

But now we deal with a module U2 such that  1 Ext (W, U2) = 0. This implies that U3 is isomorphic to U2 ⊕ W . Thus the next pushout construction yields an exact sequence of the form

0 → U → U ⊕ W → W ′ → 0. 2 2

Acknowledgement. The author is indebted to Dieter Vossieck for a careful reading of the final version of the paper.

References

[BP]    Bautista, R. and Pérez, E.: On modules and complexes without self-extensions. Communications in Algebra 34 (2006), 3139-3152.        [ Links ]

[R1]    Ringel, C.M.: Representations of k-species and bimodules. J.Algebra 41 (1976), 269-302.        [ Links ]

[R2]    Ringel, C.M.: Bricks in hereditary length categories. Resultate der Math. 6 (1983), 64-70.        [ Links ]

[R3]    Ringel, C.M.: Some algebraically compact modules I. In: Abelian Groups and Modules (ed. A. Facchini and C. Menini). Kluwer (1995), 419-439.        [ Links ]

[R4]    Ringel, C.M.: The Gabriel-Roiter measure. Bull. Sci. math. 129 (2005). 726-748.        [ Links ]

[R5]    Ringel, C.M.: Prüfer modules which are not of finite type. In preparation.        [ Links ]

[S]    Smalø, S.: Lectures on Algebras. Mar del Plata, Argentina March 2006. Revista Unión Matemática Argentina 48-2 (2007), 21-45.        [ Links ]

[Z1]    Zwara, G.: A degeneration-like order for modules. Arch. Math. 71 (1998), 437-444        [ Links ]

[Z2]    Zwara, G.: Degenerations of finite-dimensional modules are given by extensions. Compositio Mathematica 121 (2001), 205-218.        [ Links ]

 Claus Michael Ringel
Fakultät für Mathematik, Universität Bielefeld,
POBox 100 131,
D-33 501 Bielefeld, Germany
ringel@math.uni-bielefeld.de

Recibido: 5 de marzo de 2007
Aceptado: 16 de abril de 2007