The mod p cohomology of the Morava stabilizer group at large primes

TL;DR

This paper computes the mod p cohomology of the Morava stabilizer group at large primes, revealing it as an exterior algebra, using deformation techniques and a derived invariant cycles theorem.

Contribution

It introduces a deformation approach to analyze the cohomology of the Morava stabilizer group and establishes a derived invariant cycles theorem for comparison.

Findings

Cohomology is an exterior algebra on n generators.

Method involves deformations of Ravenel's Lie algebra model.

Cohomology matches the cohomology of the unitary group.

Abstract

We calculate the cohomology of the extended Morava stabilizer group of height $n$, with trivial mod $p$ coefficients, for all heights $n$ and all primes $p>>n$. The result is an exterior algebra on $n$ generators. A brief sketch of the method: we introduce a family of deformations of Ravenel's Lie algebra model $L(n,n)$ for the Morava stabilizer group scheme. This yields a family of DGAs, parameterized over an affine line and smooth except at a single point. The singular fiber is the Chevalley-Eilenberg DGA of Ravenel's Lie algebra. Consequently the cohomology of the singular fiber is the cohomology of the Morava stabilizer group, at large primes. We prove a derived version of the invariant cycles theorem from Hodge theory, which allows us to compare the cohomology of the singular fiber to the fixed-points of the Picard-Lefschetz (monodromy) operator on the cohomology of a smooth fiber. Finally, we use some new methods for constructing small models for cohomology of reductive Lie algebras to show that the cohomology of the Picard-Lefschetz fixed-points on a smooth fiber agrees with the singular cohomology $H^*(U(n);\mathbb{F}_p)$ of the unitary group, which is the desired exterior algebra.