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2020 Gorenstein-projective and semi-Gorenstein-projective modules
Claus Michael Ringel, Pu Zhang
Algebra Number Theory 14(1): 1-36 (2020). DOI: 10.2140/ant.2020.14.1

Abstract

Let A be an artin algebra. An A -module M will be said to be semi-Gorenstein-projective provided that Ext i ( M , A ) = 0 for all i 1 . All Gorenstein-projective modules are semi-Gorenstein-projective and only few and quite complicated examples of semi-Gorenstein-projective modules which are not Gorenstein-projective have been known. One of the aims of the paper is to provide conditions on A such that all semi-Gorenstein-projective left modules are Gorenstein-projective (we call such an algebra left weakly Gorenstein). In particular, we show that in case there are only finitely many isomorphism classes of indecomposable left modules which are both semi-Gorenstein-projective and torsionless, then A is left weakly Gorenstein. On the other hand, we exhibit a 6-dimensional algebra Λ with a semi-Gorenstein-projective module M which is not torsionless (thus not Gorenstein-projective). Actually, also the Λ -dual module M is semi-Gorenstein-projective. In this way, we show the independence of the total reflexivity conditions of Avramov and Martsinkovsky, thus completing a partial proof by Jorgensen and Şega. Since all the syzygy-modules of M and M are 3-dimensional, the example can be checked (and visualized) quite easily.

Citation

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Claus Michael Ringel. Pu Zhang. "Gorenstein-projective and semi-Gorenstein-projective modules." Algebra Number Theory 14 (1) 1 - 36, 2020. https://doi.org/10.2140/ant.2020.14.1

Information

Received: 6 August 2018; Revised: 22 July 2019; Accepted: 23 August 2019; Published: 2020
First available in Project Euclid: 7 April 2020

zbMATH: 07180780
MathSciNet: MR4076806
Digital Object Identifier: 10.2140/ant.2020.14.1

Subjects:
Primary: 16E65
Secondary: 16E05, 16G10, 16G50, 20G42

Rights: Copyright © 2020 Mathematical Sciences Publishers

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