On the Extension-Closed Property for the Subcategory Tr(Ω2(mod −A))

We present a monomial quiver algebra A having the property that the subcategory Tr(Ω2(mod −A)) is not extension-closed. This answers a question raised by Idun Reiten.


Introduction
For a noetherian ring , let mod − denote the category of finitely generated rightmodules.Moreover, let i (mod − ) denote the full subcategory of i-th syzygy modules of finitely generated -modules.Here a module N is an i-th syzygy module if it is a direct summand of a module of the form P ⊕ i (M) with P being projective, M a module with projective resolution Presented by: Christof Geiss Dedicated to the memory of Izzy and Leni Bernhard Böhmler bernhard.boehmler@googlemail.com René Marczinzik marczire@math.uni-bonn.deand i (M) being the kernel of f i−1 .The study of syzygies is of major importance in homologcal algebra and representation theory, we mention for example Hilbert's famous syzygy theorem [6] and the work of Auslander and Reiten on the subcategories of syzygy modules for noetherian rings in [2].Throughout, we assume that all subcategories are full.Let C denote a subcategory of mod − for a noetherian ring and let Tr denote the Auslander−Bridger transpose that gives an equivalence between the stable categories mod − and mod − op , see for example [1].Following [10], we define Tr(C) to be the smallest additive subcategory of mod − op containing the modules Tr(M) for M ∈ C and all the projective op -modules.Recall that a subcategory C is said to be closed under extensions if for any M, N ∈ C, also every module X is in C if there is an exact sequence of the form 0 → M → X → N → 0. The extension-closedness of certain subcategories can give important information on the algebra.We mention for example Theorem 0.1 in [2] which shows that the property that the subcategories i (mod − ) are extension-closed for 1 ≤ i ≤ n for an algebra is equivalent to being quasi n-Gorenstein in the sense of [8].In [10], it is noted that even if a subcategory C is not closed under extensions, the subcategory Tr(C) can still be closed under extensions.This is for example true for C = 1 (mod − ) as was remarked in [10].Idun Reiten states in the paragraph before Proposition 1.1 in [10] that the answer to the following question is not known: Question 0.1 Let be a noetherian ring and i > 1.Is Tr( i (mod − )) closed under extensions?
In this article we give a negative answer to this question.An Appendix containing the performed computer calculations is also given.Notation Throughout, let A = KQ/I be the finite-dimensional quiver algebra over a field K where Q is given by and the relations are given by I = xy, yz, zx, x 3 .
Our main result is:

The Proof
In this section we prove Theorem 0.2.We assume that the reader is familiar with the basics of representation theory and homological algebra of finite-dimensional algebras and refer for example to [3] and [11].Let A be defined as above.Let S i denote the simple A-modules, J the Jacobson radical of A, D = Hom K (−, K) the natural duality and τ = D Tr the Auslander-Reiten translate.The algebra A has vector space dimension 7 over the field K. Let e i denote the primitive idempotents of the quiver algebra A corresponding to the vertices i for i = 1, 2. The indecomposable projective A-module P 1 = e 1 A = e 1 , x, y, x 2 has dimension vector [3,1] and the indecomposable projective A-modules P 2 = e 2 A = e 2 , z, zy has dimension vector [1,2].We remark that P 2 is isomorphic to the indecomposable injective module I 2 = D(Ae 2 ).The indecomposable injective module I 1 = D(Ae 1 ) has dimension vector [3,1].We give the Auslander−Reiten quiver of A here: We remark that the algebra A is a representation-finite string algebra and thus the determination of the Auslander−Reiten quiver is well known, see for example [5].We leave the verficiation of the Auslander−Reiten quiver of A to the reader.The Auslander−Reiten quiver can also be verified using [9], since by the theory in [5], the Auslander−Reiten quiver does not depend on the field because A is a representation-finite string algebra.
The dashed arrows in the Auslander−Reiten quiver indicate the Auslander−Reiten translates and the other arrows the irreducible maps.There are 13 indecomposable A-modules and we label them with their dimension vectors and an easy description via indecomposable projectives or injectives, their radicals, simple modules, or ideals in the algebra A. For finitedimensional algebras we have D Tr = τ , the Auslander−Reiten translate.Since D is a duality, the subcategory Tr( 2 (mod −A)) is extension-closed if and only if τ ( 2 (mod −A)) is extension-closed.We will consider the subcategory τ ( 2 (mod −A)) in the following.We sometimes use the notation τ i := τ i−1 for the higher Auslander−Reiten translates that play an important role for example in the theory of cluster-tilting subcategories, see [7].Let Ĩ := (x + z)A denote the right ideal generated by x + z, which has vector space basis {x + z, x 2 , zy}.The module M 1 := A/ Ĩ is an indecomposable A-module with dimension vector [2,2].Moreover, we mention that M 1 is depicted as a quiver representation in the proof of Lemma 0.4.Let M 2 := P 2 /S 2 = e 2 A/(zy)A.This is an indecomposable A-module with dimension vector [1,1].See the Auslander−Reiten quiver of A. Lemma 0. 3 The following assertions hold: (1)

Proof
(1) We have the exact sequences It follows that 2 (M 1 ) ∼ = S 1 .We can see from the Auslander−Reiten quiver that It follows that 2 (M 2 ) ∼ = e 2 J and we can see from the Auslander−Reiten quiver that τ (e 2 J ) We have W ∼ = P 2 ⊕ U , where U = A/((x + y + z)A) is indecomposable.The short exact sequence is therefore not split exact.

The quiver representation of U looks as follows:
Note that the module U is isomorphic to the module A/(x + y + z)A and is indecomposable.A direct verification that we leave to the reader shows that W ∼ = P 2 ⊕ U .This proves that the short exact sequence is not split exact, since W is not isomorphic to the direct sum of the indecomposable modules M 1 and M 2 .
Proof We prove the equivalent statement that τ −1 (U ) is not a second syzygy module.From the Auslander−Reiten quiver we can see that τ −1 (U ) = P 1 /S 1 .Now we use the result that a general module X over an Artin algebra is a direct summand of an n-th syzygy module if and only if X is a direct summand of P ⊕ n ( −n (X)) for some projective module P , see for example Proposition 3.2 of [4].The two short exact sequences give rise to minimal injective coresolutions.Now 2 (S 1 ) ∼ = S 1 ⊕ S 2 ⊕ e 2 J 1 and 2 (M 2 ) ∼ = e 2 J 1 .But the indecomposable module P 1 /S 1 is not a direct summand of a module of the form for any projective module P .Hence, τ −1 (U ) is not a 2 nd syzygy module.

Appendix: QPA Calculations
In this section we illustrate how to verify that our result is correct over the field with three elements with the help of QPA.We maintain the notation from the previous chapter.The reader can copy and paste the following code into GAP.A hashmark indicates the beginning of a comment.
LoadPackage("qpa"); Q:=Quiver(2,[[1,1,"x"],[1,2,"y"],[2,1,"z"]]);kQ:=PathAlgebra (GF(3),Q); AssignGeneratorVariables(kQ); # in order to recognize x,y,z as elements of kQ rel:=[x * y,y * z,z * x,xˆ3];A:=kQ/rel; # define the admissible ideal Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.