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.

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Mathematical Subject Classification 2010

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