#### Vol. 14, No. 1, 2020

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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 ${Ext}^{i}\left(M,A\right)=0$ for all $i\ge 1$. 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 $\Lambda$ with a semi-Gorenstein-projective module $M$ which is not torsionless (thus not Gorenstein-projective). Actually, also the $\Lambda$-dual module ${M}^{\ast }$ 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 ${M}^{\ast }$ 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