Vol. 298, No. 2, 2019

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Tilting modules over Auslander–Gorenstein algebras

Osamu Iyama and Xiaojin Zhang

Vol. 298 (2019), No. 2, 399–416
Abstract

For a finite-dimensional algebra $\Lambda$ and a nonnegative integer $n$, we characterize when the set of additive equivalence classes of tilting modules with projective dimension at most $n$ has a minimal (or equivalently, minimum) element. This generalizes results of Happel and Unger. Moreover, for an $n$-Gorenstein algebra $\Lambda$ with $n\ge 1$, we construct a minimal element in . As a result, we give equivalent conditions for a $k$-Gorenstein algebra to be Iwanaga–Gorenstein. Moreover, for a $1$-Gorenstein algebra $\Lambda$ and its factor algebra $\Gamma =\Lambda ∕\left(e\right)$, we show that there is a bijection between and the set of additive equivalence classes of basic support $\tau$-tilting $\Gamma$-modules, where $e$ is an idempotent such that $e\Lambda$ is the additive generator of the category of projective-injective $\Lambda$-modules.

Keywords
$n$-Gorenstein algebra, tilting module, support $\tau$-tilting module
Primary: 16G10
Secondary: 16E10
Milestones
Received: 19 March 2018
Revised: 8 July 2018
Accepted: 14 July 2018
Published: 8 March 2019
Authors
 Osamu Iyama Graduate School of Mathematics Nagoya University Chikusa-ku Nagoya Japan Xiaojin Zhang School of Mathematics and Statistics Nanjing University of Information Science and Technology Nanjing China