Abstract
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.
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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
References
Auslander, M., Bridger, M.: Stable Module Theory. Memoirs of the American Mathematical Society, no. 94 American Mathematical Society, Providence, R.I. pp. 146 (1969)
Auslander, M., Reiten, I.: Syzygy Modules for Noetherian Rings. Journal of algebra 183, 167–185 (1996)
Auslander, M., Reiten, I., Smalo, S.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics, 36. Cambridge University Press, Cambridge. pp. xiv+ 425 (1997)
Auslander, M., Solberg, O.: Relative Homology and Representation Theory I. Relative Homology and Homologically Finite Subcategories. Comm. in Alg. 21(9), 2995–3031 (1993)
Butler, M., Ringel, C.: Auslander−reiten Sequences with few Middle Terms and Applications to String Algebras. Communications in Algebra 15, 145–179 (1987)
Hilbert, D.: Über die Theorie der Algebraischen Formen. Mathematische Annalen 36, 473–534 (1890)
Iyama, O.: Higher-Dimensional Auslander−Reiten Theory on Maximal Orthogonal Subcategories. Adv. Math. 210, 22–50 (2007)
Iyama, O., Zhang, X.: Tilting Modules over Auslander−Gorenstein Algebras. Pac. J. Math. 298(2), 399–416 (2019)
The QPA-team. QPA - Quivers, Path Algebras and Representations - a GAP Package, Version 1.250 (2016)
Reiten, I.: Homological Theory of Noetherian Rings. In: Representation Theory of Algebras and Related Topics. Proceedings of the Workshop at UNAM, Mexico City, Mexico, August 16-20, 1994. Providence, RI: American Mathematical Society. pp. 247–259 (1996)
Skowronski, A., Yamagata, K.: Frobenius Algebras i: Basic Representation Theory EMS Textbooks in Mathematics (2011)
Acknowledgements
The second author gratefully acknowledges funding by the DFG (with project number 428999796). We profited from the use of the GAP-package [9].
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Open Access funding enabled and organized by Projekt DEAL. Financial interests: The second author has received funding by the DFG (with project number 428999796).
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Appendix: QPA Calculations
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,x^3];A:=kQ/rel; # define the admissible ideal and the algebra projA:=IndecProjectiveModules(A); # define the PIMs of A P1:=projA[1];P2:=projA[2]; M2:=CoKernel(SocleOfModuleInclusion(P2)); I_neu:=RightIdeal(A, [A.x+A.z]); # next, transform this ideal into the module A/I_neu: O:=RightAlgebraModuleToPathAlgebraMatModule(RightAlgebraModule (A, \*, I_neu)); M1:=TransposeOfModule(NthSyzygy(TransposeOfModule(O),1)); IsomorphicModules(M1,DTr(NthSyzygy(M1,2))); IsomorphicModules(M2,DTr(NthSyzygy(M2,2))); IsIndecomposableModule(M1); IsIndecomposableModule(M2); ext:=ExtOverAlgebra(M2,M1); maps:=ext[2]; Size(maps); # compute the vector space dimension of Ext(M2,M1) monos:=List(maps,h->PushOut(ext[1],h)[1]); u:=monos[1]; IsInjective(u); IsomorphicModules(Source(u),M1); IsomorphicModules(CoKernel(u),M2); W:=Range(u); WW:=DecomposeModule(W); OO:=WW[1];IsomorphicModules(OO,P2); U:=WW[2]; IsIndecomposableModule(U); K:=TrD(U); IsNthSyzygy(K,2);
Hereby, we have constructed an injective map \(u: M_{1} \rightarrow W\), where the modules M1 and M2 := Coker(u) are indecomposable and satisfy M1≅τ3(M1) and M2≅τ3(M2). It is verified that the module W is isomorphic to P2 ⊕ U, where P2 is the indecomposable projective module corresponding to the second vertex e2 and U is some indecomposable module that is not in τ(Ω2(mod −A)), since τ− 1(U) is not a second syzygy module. The subcategory τ(Ω2(mod −A)) is therefore not extension-closed.
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Böhmler, B., Marczinzik, R. On the Extension-Closed Property for the Subcategory Tr(Ω2(mod −A)). Algebr Represent Theor 26, 1433–1440 (2023). https://doi.org/10.1007/s10468-022-10140-7
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DOI: https://doi.org/10.1007/s10468-022-10140-7