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Cohomological invariants and Brauer groups of algebraic stacks in positive characteristic

Andrea Di Lorenzo, Roberto Pirisi

Abstract

We introduce a theory of cohomological invariants with mod $p^r$ coefficients for algebraic stacks in characteristic $p$. Using these new tools we complete the computation of the Brauer group and cohomological invariants of the stack of elliptic curves over any field.

Cohomological invariants and Brauer groups of algebraic stacks in positive characteristic

Abstract

We introduce a theory of cohomological invariants with mod coefficients for algebraic stacks in characteristic . Using these new tools we complete the computation of the Brauer group and cohomological invariants of the stack of elliptic curves over any field.
Paper Structure (21 sections, 61 theorems, 271 equations)

This paper contains 21 sections, 61 theorems, 271 equations.

Table of Contents

  1. Étale motivic cohomology
  2. Invariants in characteristic $p$
  3. Invariants with coefficients in étale motivic cohomology
  4. Invariants in low degrees
  5. Étale motivic cohomology of fields
  6. Motivic cohomology, log differentials and symbols
  7. Torsion in ${\rm H}^{n+1}_{p^r}(F)$
  8. Unramified, tamely ramified and wildly ramified elements
  9. Izhboldin's description of wild ramification.
  10. Étale motivic cohomology of DVRs
  11. The case $i=j=n$
  12. The case $i=j+1=n+1$
  13. Some computations of cohomological invariants
  14. Cohomological invariants of $\mathcal{M}_{1,1}$ in positive characteristic
  15. Setup
  16. ...and 6 more sections

Key Result

Theorem 1

There is a theory of mod $p^r$ cohomological invariants for algebraic stacks in characteristic $p$ extending the classical theory, and moreover for a smooth quotient stack $\mathcal{X}$ we have:

Theorems & Definitions (129)

  • Theorem
  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • ...and 119 more