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Gorenstein-projective
and semi-Gorenstein-projective modules
Claus Michael Ringel and Pu Zhang |
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Vol. 14 (2020), No. 1, 1–36
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DOI: 10.2140/ant.2020.14.1
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Abstract |
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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. |
Keywords
Gorenstein-projective module, semi-Gorenstein-projective
module, left weakly Gorenstein algebra, torsionless module,
reflexive module, $t$-torsionfree module, Frobenius
category, $\mho$-quiver.
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Mathematical Subject Classification 2010
Primary: 16E65
Secondary: 16E05, 16G10, 16G50, 20G42
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Milestones
Received: 6 August 2018
Revised: 22 July 2019
Accepted: 23 August 2019
Published: 15 March 2020
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Authors |
| Claus Michael Ringel | |
| Fakultät für Mathematik PO Box 100131, D-33501 Universität Bielefeld Germany |
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| Pu Zhang | |
| School of Mathematical
Sciences Shanghai Jiao Tong University Shanghai 200240 P. R. China |
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