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Non-commutative crepant resolutions

Michel Van den Bergh

TL;DR

The paper introduces non-commutative crepant resolutions (NCCR) as $A=\operatorname{End}_R(M)$ with $M$ reflexive and $A$ of finite global dimension and Cohen–Macaulay over $R$, framing them as noncommutative analogues of crepant resolutions and exploring connections to the Bondal–Orlov conjecture. It proves existence of NCCR in key cases, notably three-dimensional terminal Gorenstein singularities, and provides evidence that different crepant (commutative) resolutions are derived equivalent via NCCRs, generalizing ideas from the McKay correspondence. It develops a method to construct a crepant resolution from a given NCCR by passing to moduli spaces of stable $A$-representations, establishing an equivalence between $D^b(\mathrm{coh}(Y))$ and $D^b(\mathrm{mod}(A))$ under a dimension hypothesis, and extends the framework with relative Serre duality and spanning-class techniques. The paper further applies the approach to cones over Del Pezzo surfaces and to one-dimensional torus invariants, producing explicit endomorphism algebras and tilting objects that yield NCCRs and derived equivalences in these invariant settings.

Abstract

We introduce the notion of a ``non-commutative crepant'' resolution of a singularity and show that it exists in certain cases. We also give some evidence for an extension of a conjecture by Bondal and Orlov, stating that different crepant resolutions of a Gorenstein singularity have the same derived category.

Non-commutative crepant resolutions

TL;DR

The paper introduces non-commutative crepant resolutions (NCCR) as with reflexive and of finite global dimension and Cohen–Macaulay over , framing them as noncommutative analogues of crepant resolutions and exploring connections to the Bondal–Orlov conjecture. It proves existence of NCCR in key cases, notably three-dimensional terminal Gorenstein singularities, and provides evidence that different crepant (commutative) resolutions are derived equivalent via NCCRs, generalizing ideas from the McKay correspondence. It develops a method to construct a crepant resolution from a given NCCR by passing to moduli spaces of stable -representations, establishing an equivalence between and under a dimension hypothesis, and extends the framework with relative Serre duality and spanning-class techniques. The paper further applies the approach to cones over Del Pezzo surfaces and to one-dimensional torus invariants, producing explicit endomorphism algebras and tilting objects that yield NCCRs and derived equivalences in these invariant settings.

Abstract

We introduce the notion of a ``non-commutative crepant'' resolution of a singularity and show that it exists in certain cases. We also give some evidence for an extension of a conjecture by Bondal and Orlov, stating that different crepant resolutions of a Gorenstein singularity have the same derived category.

Paper Structure

This paper contains 14 sections, 24 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. Background on homologically homogeneous rings
  4. Non-commutative crepant resolutions
  5. Resolutions with fibers of dimension $\le 1$
  6. Construction a crepant resolution starting from a non-commutative one
  7. Introduction
  8. Moduli spaces of representations
  9. Application to our situation
  10. Relative Serre duality
  11. Spanning classes
  12. Proof of the main theorem
  13. Cones over del Pezzo surfaces
  14. One-dimensional torus invariants

Key Result

Theorem 2.1

Let ${\mathcal{E}}$ be a Grothendieck category and assume that ${\mathcal{A}}=D({\mathcal{E}})$ is generated by a compact object $E$ (i.e. $E^\perp=0$ and $\operatorname {Hom}_{\mathcal{A}}(E,-)$ commutes with direct sums). Then ${\mathcal{A}}=D(\Lambda)$ where $\Lambda$ is a DG-algebra whose cohomo

Theorems & Definitions (53)

  • Example 1.1
  • Conjecture 1.2
  • Theorem 2.1
  • Example 3.1
  • Proposition 3.3
  • proof
  • Example 3.4
  • Remark 3.6
  • Definition 4.1
  • Lemma 4.2
  • ...and 43 more