Tilting modules over Auslander-Gorenstein Algebras

TL;DR

This paper characterizes when tilting modules over Auslander-Gorenstein algebras have minimal elements, constructs such elements for n-Gorenstein algebras, and explores their relation to support tau-tilting modules.

Contribution

It generalizes existing results on tilting modules, constructs minimal tilting modules for n-Gorenstein algebras, and establishes a bijection with support tau-tilting modules in certain cases.

Findings

Characterization of minimal tilting modules over Auslander-Gorenstein algebras.

Construction of minimal tilting modules for n-Gorenstein algebras.

Bijection between tilting modules and support tau-tilting modules for 1-Gorenstein algebras.

Abstract

For a finite dimensional algebra $\Lambda$ and a non-negative integer $n$, we characterize when the set $\tilt_n\Lambda$ of additive equivalence classes of tilting modules with projective dimension at most $n$ has a minimal (or equivalently, minimum) element. This generalize results of Happel-Unger. Moreover, for an $n$-Gorenstein algebra $\Lambda$ with $n\geq 1$, we construct a minimal element in $\tilt_{n}\Lambda$. As a result, we give equivalent conditions for a $k$-Gorenstein algebra to be Iwanaga-Gorenstein. Moreover, for an $1$-Gorenstein algebra $\Lambda$ and its factor algebra $\Gamma=\Lambda/(e)$, we show that there is a bijection between $\tilt_1\Lambda$ and the set $\sttilt\Gamma$ of isomorphism classes of basic support $\tau$-tilting $\Gamma$-modules, where $e$ is an idempotent such that $e\Lambda $ is the additive generator of projective-injective $\Lambda$-modules.