Gorenstein-projective and semi-Gorenstein-projective modules

Ringel, Claus Michael; Zhang, Pu

doi:10.2140/ant.2020.14.1

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Form

Policies for Authors

Ethics Statement

ISSN: 1944-7833 (e-only)

ISSN: 1937-0652 (print)

Author Index

To Appear

Other MSP Journals

This article is available for purchase or by subscription. See below.

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.

PDF Access Denied

We have not been able to recognize your IP address 3.128.199.162 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

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