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On de Rham--Witt Cohomology of Classifying Stacks

Shizhang Li, Yuan Yang

Abstract

We give an example of proper smooth fourfold over a perfect field k of characteristic p > 0 with asymmetric Hodge--Witt numbers in total degree 3. Our example is sharp both in terms of dimension and total degree. We arrive at our example by computing and approximating the Hodge--Witt cohomology groups of the classifying stack B alpha_p.

On de Rham--Witt Cohomology of Classifying Stacks

Abstract

We give an example of proper smooth fourfold over a perfect field k of characteristic p > 0 with asymmetric Hodge--Witt numbers in total degree 3. Our example is sharp both in terms of dimension and total degree. We arrive at our example by computing and approximating the Hodge--Witt cohomology groups of the classifying stack B alpha_p.

Paper Structure

This paper contains 12 sections, 69 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Left graded $\mathscr{R}$-modules
  3. The ring $\mathscr{R}$ and totalization
  4. Coherent $\mathscr{R}$-modules
  5. Ekedahl's completion functor
  6. Hodge--Witt numbers
  7. Deeper structures on $\mathscr{R}$-complexes
  8. Duality
  9. Ekedahl's star product
  10. Diagonal $t$-structure
  11. de Rham--Witt cohomology of smooth stacks
  12. de Rham--Witt cohomology of $B\alpha_p$

Key Result

Theorem 1.1

There exists a smooth proper $4$-fold over any perfect field of characteristic $p > 0$, with $h_W^{0,3} = 1$, $h_W^{1,2} = -2$, $h_W^{2,1} = 1$, and $h_W^{3,0} = 0$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (151)

  • Theorem 1.1: \ref{['the counterexample']}
  • Definition 2.1
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Corollary 2.7
  • Remark 2.9
  • Theorem 2.10
  • proof
  • Proposition 2.11
  • ...and 141 more