Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Mohammad Behzad Kang, Andrew Salch

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.

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

Abstract

We calculate the cohomology of the extended Morava stabilizer group of height , with trivial mod coefficients, for all heights and all primes . The result is an exterior algebra on generators. A brief sketch of the method: we introduce a family of deformations of Ravenel's Lie algebra model 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 of the unitary group, which is the desired exterior algebra.

Paper Structure

This paper contains 26 sections, 31 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. What this paper is about
  3. The reduction to Lie algebra cohomology
  4. Deformations of Ravenel's Lie algebra model for the Morava stabilizer group
  5. Outline of proof of the main theorem
  6. Acknowledgments and conventions
  7. Models for Lie algebra cohomology
  8. Derivations and chain homotopy retractions on Chevalley-Eilenberg DGAs
  9. Some elementary linear algebra
  10. How to build a model of a DGA from a derivation on that DGA
  11. Small models for the cohomology of a Lie algebra
  12. The critical complex in $\Lambda^{\bullet}_k(\mathfrak{gl}_n(k)^*)$ models the modular Lie algebra $\mathfrak{gl}_n(k)$
  13. The descent problem for $H^*(U(n);k)$
  14. Monodromy and invariant cycles in a family of deformations of Ravenel's model
  15. Deformations of Ravenel's Lie algebra model for the Morava stabilizer group
  16. ...and 11 more sections

Key Result

Theorem A

Let $n$ be a positive integer. Then, for all sufficiently large primes $p$, $H^*\left(\mathop{\mathrm{{\bf G}_n}}\nolimits; \mathbb{F}_{p^n}\right)$ is isomorphic, as a graded $\mathbb{F}_p$-algebra, to the associated graded of a finite filtration on the singular cohomology $H^*(U(n);\mathbb{F}_p)$

Theorems & Definitions (82)

  • Theorem A
  • Definition 2.2.1
  • Remark 2.2.2
  • Lemma 2.2.3
  • proof
  • Definition 2.2.4
  • Example 2.2.5
  • Lemma 2.2.6
  • proof
  • Proposition 2.2.7
  • ...and 72 more