Totaro's inequality for classifying spaces

TL;DR

This paper discusses Totaro's conjecture relating the cohomology of classifying spaces and stacks for complex Lie groups in characteristic p, providing a shorter proof of a recent result.

Contribution

It offers a more concise proof of Totaro's conjecture on the cohomology bounds of classifying spaces and stacks for complex Lie groups in characteristic p.

Findings

Confirmed Totaro's conjecture with a shorter proof

Bounded the dimension of singular cohomology by de Rham cohomology

Validated recent results by Kubrak--Prikhodko

Abstract

For a complex Lie group G and a prime number p, Totaro had conjectured that the dimension of the singular cohomology with Z/p-coefficients of classifying space of G is bounded above by that of the de Rham cohomology of the classifying stack of (the split form of) G in characteristic p. This conjecture was recently proven by Kubrak--Prikhodko. In this note, we give a shorter proof.