Tilting theory for hypersurface singularities of dimension one
Osamu Iyama, Junyang Liu
TL;DR
The article extends tilting theory for $ obreak$N-graded Gorenstein hypersurface singularities by giving explicit presentations for the endomorphism algebra of the standard silting object $V$ in the stable category $\underline{\mathrm{CM}}_0^{\mathbb{Z}}R$. When the $a$-invariant is non-negative, the endomorphism algebra $\Gamma$ is finite-dimensional and Iwanaga-Gorenstein with self-injective dimension at most $2$, and is presented as $\Gamma\cong kQ/I_{r,s}$ with a concrete surjection from $kQ$ whose kernel is $I_{r,s}$; for negative $a$, the dg endomorphism algebra is Gorenstein and modeled by a dg path algebra $(kQ',d)$, linking to a triangle equivalence with $\mathrm{per}(kQ',d)$. The paper covers numerical semigroup algebras as a primary application and supplies Auslander–Reiten quivers for finite and countable Cohen–Macaulay types, locating the standard silting object within them. A Serre-functor criterion for Gorensteinness of homologically finite dg algebras underpins the broader framework, unifying the non-negative and negative $a$-invariant cases. Collectively, these results generalize and sharpen previous work by Buchweitz–Iyama–Yamaura to hypersurface settings with non-standard gradings and non-algebraically closed base fields, and provide explicit algebraic and dg-model descriptions useful for computations and SEO-oriented indexing.
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.