Tilting theory for hypersurface singularities of dimension one

Abstract

Any $\mathbb{N}$-graded commutative Gorenstein ring $R$ of Krull dimension one with $R_0$ a field admits a standard silting object $V$ in the stable category $\underline{\mathrm{CM}}_0^{\mathbb{Z}}R$, and the object $V$ is tilting if and only if the $a$-invariant $a$ is non-negative, as shown by Buchweitz, the first author, and Yamaura. In this article, under the additional assumption that $R$ is a hypersurface singularity, we prove that endomorphism algebra of $V$ is Iwanaga-Gorenstein of self-injective dimension at most $2$, and we give its explicit presentation in terms of a quiver with relation. In the case of where $a$ is negative, we prove that the dg endomorphism algebra of $V$ is Gorenstein, and we give its explicit presentation in terms of a dg path algebra. We explain our results by several examples including numerical semigroup algebras generated by two elements. Moreover, for each finite and countable Cohen-Macaulay representation type, we include the Auslander-Reiten quiver of the category $\mathrm{CM}_0^{\mathbb{Z}}R$ with the position of the standard silting object. As a step of the proof of our results, we give a characterization of Gorensteinness of homologically finite dg algebras in terms of Serre functors.