Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Norihiro Hanihara

TL;DR

We address the relations between graded and ungraded singularity categories of toric Gorenstein algebras (Veronese subrings and Segre products) and higher representation theory. By constructing tilting objects in the graded singularity category with endomorphism algebras that are $(d-a-1)$-representation infinite and carrying strict $a$-th root pairs, we obtain equivalences with folded cluster categories and realize Calabi–Yau completions as twisted CY algebras, described explicitly via quivers and relations. The work demonstrates preservation of higher representation infiniteness under certain idempotent quotients and provides explicit quiver presentations for the resulting algebras, along with Orlov-type connections between graded singularities and non-commutative projective geometry. In the Segre-product case, a detailed development yields a hereditary instance at $n=3$ and a commutative diagram of equivalences that respects 3-cluster tilting structures, illustrating the broader applicability of the folding and tilting framework to toric singularities.

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.

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

TL;DR

We address the relations between graded and ungraded singularity categories of toric Gorenstein algebras (Veronese subrings and Segre products) and higher representation theory. By constructing tilting objects in the graded singularity category with endomorphism algebras that are -representation infinite and carrying strict -th root pairs, we obtain equivalences with folded cluster categories and realize Calabi–Yau completions as twisted CY algebras, described explicitly via quivers and relations. The work demonstrates preservation of higher representation infiniteness under certain idempotent quotients and provides explicit quiver presentations for the resulting algebras, along with Orlov-type connections between graded singularities and non-commutative projective geometry. In the Segre-product case, a detailed development yields a hereditary instance at and a commutative diagram of equivalences that respects 3-cluster tilting structures, illustrating the broader applicability of the folding and tilting framework to toric singularities.

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.

Paper Structure

This paper contains 12 sections, 29 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Results
  4. Reduction of higher representation infinite algebras
  5. Folded cluster categories and strict root pairs
  6. Veronese subrings
  7. Tilting theory
  8. Quivers and relations
  9. Homogeneous coordinate ring of a product of projective lines
  10. Derived Gorenstein parameters and Orlov's theorem
  11. Reduction of $n$-representation infinite algebras and proof of \ref{['P^1']}
  12. Proof of \ref{['n=3']}

Key Result

Theorem 1.1

Let $k$ be an arbitrary field, $a\geq1$ and $n\geq2$ be arbitrary integers, $d=an$, and $S=k[x_1,\ldots,x_d]$. Let $R=S^{(n)}$ be the $n$-th Veronese subalgebra. Give a grading on the $R$-module $S$ (in particular on the ring $R$) by requiring a monimial of degree $cn+j$ to have degree $c$ when $0\l Then we have the following.

Theorems & Definitions (62)

  • Theorem 1.1: =\ref{['VER']}
  • Theorem 1.2: =\ref{['qgr']}
  • Theorem 1.3: =\ref{['P^1']}, \ref{['n=3']}
  • Theorem 1.4: =\ref{['-1RI']}
  • Definition 2.1: HIO
  • Theorem 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 52 more