Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ind-geometric stacks

Sabin Cautis, Harold Williams

Abstract

We develop the theory of ind-geometric stacks, in particular their coherent and ind-coherent sheaf theory. This provides a convenient framework for working with equivariant sheaves on ind-schemes, especially in derived settings. Motivating examples include the coherent Satake category, the double affine Hecke category, and related categories in the theory of Coulomb branches.

Ind-geometric stacks

Abstract

We develop the theory of ind-geometric stacks, in particular their coherent and ind-coherent sheaf theory. This provides a convenient framework for working with equivariant sheaves on ind-schemes, especially in derived settings. Motivating examples include the coherent Satake category, the double affine Hecke category, and related categories in the theory of Coulomb branches.
Paper Structure (36 sections, 92 theorems, 70 equations)

This paper contains 36 sections, 92 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Technical overview
  3. Relations to existing literature
  4. Conventions
  5. Geometric stacks
  6. Definitions
  7. Truncated and classical geometric stacks
  8. Coherent sheaves
  9. Pushforward and base change
  10. Cohomological dimension
  11. Proper and almost finitely presented morphisms
  12. Closed immersions
  13. Convergence
  14. Ind-geometric stacks
  15. Definitions
  16. ...and 21 more sections

Key Result

Proposition 3.2

Geometric morphisms are stable under composition and base change in $\mathrm{Stk}_k$. If $f: X \to Y$ is a morphism in $\mathrm{Stk}_k$, then $f$ is geometric if $X$ and $Y$ are, and $X$ is geometric if $f$ and $Y$ are. In particular, $\mathrm{GStk}_{k}$ is closed under fiber products in $\mathrm{St

Theorems & Definitions (181)

  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Proposition 3.6
  • proof
  • Proposition 3.7
  • Proposition 3.8
  • ...and 171 more