Tilting modules over Auslander-Gorenstein Algebras

Abstract

For a finite dimensional algebra $\Lambda$ and a non-negative integer $n$, we characterize when the set $\tilt_nΛ$ 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 $n\ge 1$, we construct a minimal element in $\tilt_{n}Λ$. 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 $\tilt_1Λ$ and the set $\sttiltΓ$ 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.

References in corpus (2)

• $\tau$-tilting finite algebras, bricks and $g$-vectors
• Dominant dimensions, derived equivalences and tilting modules

Cited by in corpus (2)

• Classifying $\tau$-tilting modules over the Auslander algebra of $K[x]/(x^n)$
• Tilting modules and dominant dimension with respect to injective modules