Algebra & Number Theory
- Algebra Number Theory
- Volume 14, Number 1 (2020), 1-36.
Gorenstein-projective and semi-Gorenstein-projective modules
Let be an artin algebra. An -module will be said to be semi-Gorenstein-projective provided that for all . 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 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 is left weakly Gorenstein. On the other hand, we exhibit a 6-dimensional algebra with a semi-Gorenstein-projective module which is not torsionless (thus not Gorenstein-projective). Actually, also the -dual module 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 and are 3-dimensional, the example can be checked (and visualized) quite easily.
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
Permanent link to this document
Digital Object Identifier
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]
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. https://projecteuclid.org/euclid.ant/1586224818