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Non-commutative deformations of tilting bundles and the derived McKay correspondence

Yujiro Kawamata

TL;DR

The paper develops a framework to extend tilting bundles and the derived McKay correspondence to formal non-commutative (NC) deformations of algebraic varieties. By constructing Čech-type resolutions on NC schemes, it controls extensions of locally free modules and tilting bundles, establishing obstruction theories and cohomological criteria that permit lifting to NC deformations. It proves that, starting from a tilting bundle $F^0$ on $X^0$, there exists a unique tilting bundle $F$ on any NC deformation $A$ over an Artin local ring, with endomorphism algebra $E$ that yields a derived equivalence $D^b\text{coh}(A) \simeq D^b\text{mod-}E$, extending the Bondal–Rickard framework to the NC setting. The results show that when the torension-free endomorphism algebra and tilting conditions hold, the derived McKay correspondence persists under NC deformations, with $E$ remaining noetherian through an induction on the deformation length. This work thus broadens the applicability of tilting-theoretic and McKay-type correspondences to non-commutative geometric contexts.

Abstract

We prove that the tilting bundle and the derived McKay correspondence extends under formal non-commutative deformations by using Cech cohomology of non-commutative schemes.

Non-commutative deformations of tilting bundles and the derived McKay correspondence

TL;DR

The paper develops a framework to extend tilting bundles and the derived McKay correspondence to formal non-commutative (NC) deformations of algebraic varieties. By constructing Čech-type resolutions on NC schemes, it controls extensions of locally free modules and tilting bundles, establishing obstruction theories and cohomological criteria that permit lifting to NC deformations. It proves that, starting from a tilting bundle on , there exists a unique tilting bundle on any NC deformation over an Artin local ring, with endomorphism algebra that yields a derived equivalence , extending the Bondal–Rickard framework to the NC setting. The results show that when the torension-free endomorphism algebra and tilting conditions hold, the derived McKay correspondence persists under NC deformations, with remaining noetherian through an induction on the deformation length. This work thus broadens the applicability of tilting-theoretic and McKay-type correspondences to non-commutative geometric contexts.

Abstract

We prove that the tilting bundle and the derived McKay correspondence extends under formal non-commutative deformations by using Cech cohomology of non-commutative schemes.
Paper Structure (6 sections, 12 theorems, 33 equations)

This paper contains 6 sections, 12 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. NC scheme
  3. NC deformation
  4. Čech cohomology
  5. extension of locally free modules and tilting bundles
  6. derived McKay correspondence

Key Result

Theorem 1.1

Let $f^0: X^0 \to \text{Spec}(B^0)$ be a proper morphism from an algebraic variety to an affine variety, and let $F^0$ be a tilting bundle on $X^0$. Let $(R, \frak m)$ be an Artin local $k$-algebra with $R/\frak m = k$, and let $A = (A_i,\phi_{ij})$ be an NC deformation of $X^0$ over $R$. Then there given by $\Phi(x) = R\text{Hom}_A(F, x)$ with its quasi-inverse given by $\Psi(y) = y \otimes^{L}_E

Theorems & Definitions (27)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 17 more