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 Exti(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.

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