Eilenberg-Moore spectral sequence and Hodge cohomology of classifying stacks

TL;DR

This paper develops spectral sequences to compute Hodge and de Rham cohomology of classifying stacks of certain algebraic groups, revealing deep connections with topological cohomology and providing explicit algebraic descriptions.

Contribution

It constructs Eilenberg-Moore-type spectral sequences for classifying stacks and relates their cohomology to classical topological invariants, extending previous work to new algebraic groups over finite fields.

Findings

Spectral sequences converge to Hodge and de Rham cohomology of classifying stacks.

Hodge and de Rham cohomology algebras are isomorphic to singular cohomology of classifying spaces.

Explicit descriptions of certain cohomology groups over _2 are obtained.

Abstract

Let $G$ be a smooth connected reductive group over a field $k$ and $\Gamma$ be a central subgroup of $G$. We construct Eilenberg-Moore-type spectral sequences converging to the Hodge and de Rham cohomology of $B(G/\Gamma)$. As an application, building upon work of Toda and using Totaro's inequality, we show that for all $m\geq 0$ the Hodge and de Rham cohomology algebras of the classifying stacks $B\mathrm{PGL}_{4m+2}$ and $B\mathrm{PSO}_{4m+2}$ over $\mathbb{F}_2$ are isomorphic to the singular $\mathbb{F}_2$-cohomology of the classifying space of the corresponding Lie group. From this we obtain a full description of $H^{>0}(\mathrm{GL}_{4m+2}, \operatorname{Sym}^j(\mathfrak{pgl}_{4m+2}^\vee))$ and $H^{>0}(\mathrm{SO}_{4m+2}, \operatorname{Sym}^j(\mathfrak{pso}_{4m+2}^\vee))$ over $\mathbb{F}_2$.