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Perfect complexes on finite flat affine groupoids

Eike Lau

Abstract

We compute the Balmer spectrum of the category of perfect complexes on an algebraic stack admitting a finite locally free cover by an affine scheme and identify it with the homogeneous spectrum of the cohomology ring.

Perfect complexes on finite flat affine groupoids

Abstract

We compute the Balmer spectrum of the category of perfect complexes on an algebraic stack admitting a finite locally free cover by an affine scheme and identify it with the homogeneous spectrum of the cohomology ring.
Paper Structure (36 sections, 40 theorems, 26 equations)

This paper contains 36 sections, 40 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tensor triangular geometry, comparison map
  4. Sheaves of modules
  5. Algebraic stacks
  6. Affine groupoids
  7. Quasi-coherent sheaves and comodules
  8. Derived categories of sheaves
  9. Cohomology of quasi-coherent sheaves
  10. Quotient case
  11. Resolution property
  12. Finite locally free affine groupoids
  13. Well-nigh affine stacks
  14. Coarse moduli space
  15. Quotient presentation
  16. ...and 21 more sections

Key Result

Theorem A

If the algebraic stack $\mathcal{X}$ admits a finite locally free covering by an affine scheme, then $\rho_\mathcal{X}$ is a homeomorphism.

Theorems & Definitions (97)

  • Theorem A
  • Lemma B
  • Lemma C
  • Lemma D
  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.6
  • proof
  • Corollary 3.7
  • proof
  • ...and 87 more