Tilting modules over Auslander–Gorenstein algebras

Osamu Iyama and Xiaojin Zhang

Vol. 298 (2019), No. 2, 399–416

DOI: 10.2140/pjm.2019.298.399

Abstract

For a finite-dimensional algebra $\Lambda$ and a nonnegative integer $n$, we characterize when the set ${\text{ tilt}}_n\Lambda$ 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 ${\text{ tilt}}_n\Lambda$. 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 ${\text{ tilt}}_1\Lambda$ and the set $\text{ s}\tau \text{ -tilt }\Gamma$ 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

Mathematical Subject Classification 2010

Primary: 16G10

Secondary: 16E10

Milestones

Received: 19 March 2018

Revised: 8 July 2018

Accepted: 14 July 2018

Published: 8 March 2019