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On the structure of equivariant derived categories

Daniel Halpern-Leistner

Abstract

In this expository note, we discuss some results of the author on the structure of derived categories of equivariant coherent sheaves and the derived categories of geometric invariant theory quotients. We take a recent perspective, emphasizing the theory of restricted local cohomology. We also discuss several applications and concrete examples: studying the effects of birational modification on derived categories, constructing categorical completions of equivariant derived categories, and constructing actions of generalized braid groups on derived categories of GIT quotients. This is a contribution to the proceedings of the International Congress of Basic Science, held in July 2024.

On the structure of equivariant derived categories

Abstract

In this expository note, we discuss some results of the author on the structure of derived categories of equivariant coherent sheaves and the derived categories of geometric invariant theory quotients. We take a recent perspective, emphasizing the theory of restricted local cohomology. We also discuss several applications and concrete examples: studying the effects of birational modification on derived categories, constructing categorical completions of equivariant derived categories, and constructing actions of generalized braid groups on derived categories of GIT quotients. This is a contribution to the proceedings of the International Congress of Basic Science, held in July 2024.

Paper Structure

This paper contains 9 sections, 5 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. The origin and history
  3. Restricted local cohomology
  4. Semiorthogonal decompositions
  5. The example of A1/Gm
  6. Restricted cohomology in general
  7. Application: wall crossing for derived categories
  8. Application: categorical localization
  9. Application: generalized braid group actions

Key Result

Theorem 2.1

The inclusion $\mathop{\mathrm{D}}\nolimits^{\rm qc}_S(\mathbb{A}^1/\mathbb{G}_m)_{\geq w} \subset \mathop{\mathrm{D}}\nolimits^{\rm qc}(\mathbb{A}^1/\mathbb{G}_m)$ admits a right adjoint $R\Gamma_S^{\geq w}(-)$ that agrees with E:restricted_local_cohomology_example, and $R\Gamma_S^{\geq w}$ preserv

Theorems & Definitions (8)

  • Theorem 2.1
  • proof : Sketch of proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Example 4.1
  • Theorem 5.1
  • Theorem 5.2