Gorenstein-projective and semi-Gorenstein-projective modules

Ringel, Claus Michael; Zhang, Pu

doi:10.2140/ant.2020.14.1

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Gorenstein-projective and semi-Gorenstein-projective modules

Claus Michael Ringel and Pu Zhang

Vol. 14 (2020), No. 1, 1–36

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 $i \geq 1$ . 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.

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Keywords

Gorenstein-projective module, semi-Gorenstein-projective module, left weakly Gorenstein algebra, torsionless module, reflexive module, $t$-torsionfree module, Frobenius category, $\mho$-quiver.

Mathematical Subject Classification 2010

Primary: 16E65

Secondary: 16E05, 16G10, 16G50, 20G42

Milestones

Received: 6 August 2018

Revised: 22 July 2019

Accepted: 23 August 2019

Published: 15 March 2020

Authors

Claus Michael Ringel
Fakultät für Mathematik PO Box 100131, D-33501 Universität Bielefeld Germany

Pu Zhang
School of Mathematical Sciences Shanghai Jiao Tong University Shanghai 200240 P. R. China

Vol. 14, No. 1, 2020