Tilting modules over Auslander-Gorenstein Algebras

For a finite dimensional algebra $\mathrm{\Lambda }$ and a non-negative integer $n$, we characterize when the set ${\text{tilt}}_n\mathrm{\Lambda }$ 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 $\mathrm{\Lambda }$ with $n\ge 1$, we construct a minimal element in ${\text{tilt}}_n\mathrm{\Lambda }$. As a result, we give equivalent conditions for a $k$-Gorenstein algebra to be Iwanaga-Gorenstein. Moreover, for an $1$-Gorenstein algebra $\mathrm{\Lambda }$ and its factor algebra $\mathrm{\Gamma }=\mathrm{\Lambda }\mathrm{/}(e)$, we show that there is a bijection between ${\text{tilt}}_1\mathrm{\Lambda }$ and the set $\text{sttilt}\mathrm{\Gamma }$ of isomorphism classes of basic support $\tau$-tilting $\mathrm{\Gamma }$-modules, where $e$ is an idempotent such that $e\mathrm{\Lambda }$ is the additive generator of projective-injective $\mathrm{\Lambda }$-modules.