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Non-commutative crepant resolutions of toric singularities with divisor class group of rank one

Ryu Tomonaga

TL;DR

For Gorenstein toric singularities R with rk Cl(R)=1, the paper proves existence and a complete classification of toric NCCRs by translating the problem into a combinatorial upper-set framework. A bijection is established between non-trivial upper sets in a quotient group H (built from the divisor class data) and toric NCCRs given by M_J=⊕_{g∈J} S_g, with explicit maps I↦q^{-1}(J(I)). It then shows that Iyama–Wemyss mutations correspond to mutations of upper sets, implying all toric NCCRs are connected by iterated IW mutations and hence are derived equivalent. The work extends to cases with torsion in Cl(R) and provides detailed examples, including new NCCRs beyond prior constructions, thereby deepening the link between toric geometry, CM representation theory, and mutation theory.

Abstract

We prove the existence and give a classification of toric non-commutative crepant resolutions (NCCRs) of Gorenstein toric singularities with divisor class group of rank one. We prove that they correspond bijectively to non-trivial upper sets in a certain quotient of the divisor class group equipped with a certain partial order. By this classification, we show that for such toric singularities, all toric NCCRs are connected by iterated Iyama-Wemyss mutations.

Non-commutative crepant resolutions of toric singularities with divisor class group of rank one

TL;DR

For Gorenstein toric singularities R with rk Cl(R)=1, the paper proves existence and a complete classification of toric NCCRs by translating the problem into a combinatorial upper-set framework. A bijection is established between non-trivial upper sets in a quotient group H (built from the divisor class data) and toric NCCRs given by M_J=⊕_{g∈J} S_g, with explicit maps I↦q^{-1}(J(I)). It then shows that Iyama–Wemyss mutations correspond to mutations of upper sets, implying all toric NCCRs are connected by iterated IW mutations and hence are derived equivalent. The work extends to cases with torsion in Cl(R) and provides detailed examples, including new NCCRs beyond prior constructions, thereby deepening the link between toric geometry, CM representation theory, and mutation theory.

Abstract

We prove the existence and give a classification of toric non-commutative crepant resolutions (NCCRs) of Gorenstein toric singularities with divisor class group of rank one. We prove that they correspond bijectively to non-trivial upper sets in a certain quotient of the divisor class group equipped with a certain partial order. By this classification, we show that for such toric singularities, all toric NCCRs are connected by iterated Iyama-Wemyss mutations.

Paper Structure

This paper contains 11 sections, 26 theorems, 46 equations.

Table of Contents

  1. Combinatorics on upper sets
  2. Preliminaries on toric singularities
  3. Divisorial modules
  4. Relations among $G$, $\sigma$ and $R$
  5. Preliminaries on NCCRs
  6. NCCRs of toric singularities with divisor class group of rank one
  7. When are divisorial modules Cohen-Macaulay?
  8. NCCRs
  9. Iyama-Wemyss mutations of toric NCCRs
  10. Examples
  11. NCCRs of toric singularities with torsion divisor class groups

Key Result

Theorem 1

(Theorem upNCCRcorr) Let $R$ be a Gorenstein toric singularity with $\mathop{\mathrm{rk}}\nolimits\mathop{\mathrm{Cl}}\nolimits(R)=1$. In the above notations, we have a bijection between the following sets. A bijection from (1) to (2) is given by $I\mapsto q^{-1}(J(I))$ where $J(I):=I\cap(I^c+p)\subseteq H$.

Theorems & Definitions (56)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Definition 1.1
  • Lemma 1.2
  • proof
  • Theorem 1.3
  • Proposition 1.4
  • proof
  • Proposition 1.5
  • ...and 46 more