Tilting modules over Auslander-Gorenstein Algebras

For a finite dimensional algebra $Λ$ 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 $Λ$ with $n\geq 1$, 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 $Λ$ and its factor algebra $Γ=Λ/(e)$, we show that there is a bijection between $\tilt_1Λ$ and the set $\sttiltΓ$ of isomorphism classes of basic support $τ$-tilting $Γ$-modules, where $e$ is an idempotent such that $eΛ$ is the additive generator of projective-injective $Λ$-modules.

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