Higher hereditary algebras and Calabi-Yau algebras arising from some toric singularities

TL;DR

This paper explores the relationship between singularity categories of certain toric Gorenstein singularities and Calabi-Yau algebras, introducing new tilting objects and establishing categorical equivalences.

Contribution

It constructs specific tilting objects with higher representation infinite endomorphism rings and links singularity categories to cluster and Calabi-Yau algebras.

Findings

Existence of tilting objects with higher representation infinite endomorphism rings

Equivalences between singularity categories and cluster categories

Explicit descriptions of twisted Calabi-Yau algebras and higher representation infinite algebras

Abstract

We study graded and ungraded singularity categories of some commutative Gorenstein toric singularities, namely, Veronese subrings of polynomial rings, and Segre products of some copies of polynomial rings. We show that the graded singularity category has a tilting object whose endomorphism ring is higher representation infinite. Moreover, we construct the tilting object so that the endomorphism ring has a strict root pair of its higher Auslander-Reiten translation, which allows us to give equivalences between singularity categories and (folded) cluster categories in a such a way that their cluster tilting objects correspond to each other. Our distinguished form of tilting objects also allows us to construct (twisted) Calabi-Yau algebras as the Calabi-Yau completions of the root pairs. We give an explicit description of these twisted Calabi-Yau algebras as well as the higher representation infinite algebras in terms of quivers and relations. Along the way, we prove that certain idempotent quotients of higher representation infinite algebras remain higher representation infinite.