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On the Equivariant Derived Category of Perverse Sheaves

Geoff Vooys

Abstract

In this paper we extend Beilinson's realization formalism for triangulated categories and filtered triangulated categories to a pseudofunctorial and pseudonatural setting. As a consequence we prove an equivariant version of Beilinson's Theorem: for any algebraic group $G$ over an algebraically closed field $K$ and for any $G$-variety $X$, there is an equivalence of categories $D_G^b(X; \overline{\mathbb{Q}}_{\ell}) \simeq D_G^b(\mathbf{Perv}(X;\overline{\mathbb{Q}}_{\ell}))$ where $\ell$ is an integer prime coprime to the characteristic of $K$. We also show that the equivariant analogues of the other non-$D$-module aspects of Beilinson's Theorem hold in the equivariant case.

On the Equivariant Derived Category of Perverse Sheaves

Abstract

In this paper we extend Beilinson's realization formalism for triangulated categories and filtered triangulated categories to a pseudofunctorial and pseudonatural setting. As a consequence we prove an equivariant version of Beilinson's Theorem: for any algebraic group over an algebraically closed field and for any -variety , there is an equivalence of categories where is an integer prime coprime to the characteristic of . We also show that the equivariant analogues of the other non--module aspects of Beilinson's Theorem hold in the equivariant case.
Paper Structure (6 sections, 29 theorems, 72 equations)

This paper contains 6 sections, 29 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Main Results and Structure of the Paper
  3. Acknowledgments
  4. Equivariant Categories, Descent, and Pseudocones
  5. Pseudocone Realization Functors
  6. Equivariant Beilinson's Theorem and Ext Functors

Key Result

Theorem 1

The following equivalences of categories hold:

Theorems & Definitions (76)

  • Theorem 1: Cf. Theorem \ref{['Thm: Equivariant Beilinson']}
  • Corollary 2: Cf. Corollary \ref{['Cor: Section Equiv Beilin: EDC is EDPer for all var']}
  • Theorem 3: Cf. Theorem \ref{['Thm: Section Pseudocone Realizations: Pseudofunctor Realization']}
  • Theorem 4: Cf. Theorem \ref{['Thm: Section Pseudocone Realization: Pseudocone Realization']}
  • Proposition 5: Cf. Proposition \ref{['Prop: The Exty boi']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3: LusztigCuspidal2
  • Remark 2.4
  • Definition 2.5: LusztigCuspidal2
  • ...and 66 more