Algebra & Number Theory

Gorenstein-projective and semi-Gorenstein-projective modules

Claus Michael Ringel and Pu Zhang

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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.

Article information

Algebra Number Theory, Volume 14, Number 1 (2020), 1-36.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16E65: Homological conditions on rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
Secondary: 16E05: Syzygies, resolutions, complexes 16G10: Representations of Artinian rings 16G50: Cohen-Macaulay modules 20G42: Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50]

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


Ringel, Claus Michael; Zhang, Pu. Gorenstein-projective and semi-Gorenstein-projective modules. Algebra Number Theory 14 (2020), no. 1, 1--36. doi:10.2140/ant.2020.14.1.

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