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Non-Commutative Deformations of Derived McKay Correspondence for A(n) singularities

Yujiro Kawamata

Abstract

The derived McKay correspondence conjecture says that there is an equivalence of triangulated categories between the bounded derived categories of commutative and non-commutative crepant resolutions of a Gorenstein singularity. We will prove that this derived equivalence extends between the semi-universal non-commutative deformations of the commutative and the non-commutative crepant resolutions of a toric surface singularity.

Non-Commutative Deformations of Derived McKay Correspondence for A(n) singularities

Abstract

The derived McKay correspondence conjecture says that there is an equivalence of triangulated categories between the bounded derived categories of commutative and non-commutative crepant resolutions of a Gorenstein singularity. We will prove that this derived equivalence extends between the semi-universal non-commutative deformations of the commutative and the non-commutative crepant resolutions of a toric surface singularity.

Paper Structure

This paper contains 14 sections, 32 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. NC schemes and (quasi-)coherent sheaves
  3. NC deformation of CCR and NCCR for $A_n$
  4. $\text{Qcoh}(\mathcal{X})$ for $A_n$
  5. Čech cohomology
  6. Generator
  7. Comparison of two cohomology groups
  8. Comparison of $\text{End}(\mathcal{T})$ and $\mathcal{S}$
  9. Decomposition of $\mathcal{S}$ by orthogonal idempotents
  10. Decomposition of $\mathcal{A}$ by orthogonal idempotents
  11. Algebra homomorphism $\mathcal{S} \to \mathcal{A}$
  12. Injectivity of $\phi$
  13. Surjectivity of $\phi$
  14. Derived equivalence

Key Result

Theorem 1.1

Let $Y$ be a $2$-dimensional toric singularity of type $A_n$, let $X$ and $S$ be its CCR and NCCR, and let $\mathcal{X}$ and $\mathcal{S}$ be their semi-universal NC deformations over polynomial algebras $k[t_0,\dots,t_n]$ and $k[s_0,\dots,s_n]$ respectively. Then there is an equivalence of triangul which is linear over a linear coordinate change $(t_0,\dots,t_n) \to (s_0,\dots,s_n)$.

Theorems & Definitions (64)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • Definition 4.2
  • ...and 54 more