# Chromatic (co)homology of finite general linear groups

## TL;DR

This paper investigates the Morava $E$-theory of classifying spaces of finite general linear groups over finite fields, revealing a polynomial ring structure and detailed algebraic properties.

## Contribution

It establishes that the Morava $E$-theory of $BGL_d(F)$ forms a polynomial ring under one product and describes the induced structure under another, providing new algebraic insights.

## Key findings

- The Morava $E$-theory forms a polynomial ring under the product $ imes$.
- The module of $ imes$-indecomposables has a well-described $ullet$-product structure.
- Numerous auxiliary structural results about the algebraic framework were proved.

## Abstract

We study the Morava $E$-theory (at a prime $p$) of $BGL_d(F)$, where $F$ is a finite field with $|F|=1\pmod{p}$. Taking all $d$ together, we obtain a structure with two products $\times$ and $\bullet$. We prove that it is a polynomial ring under $\times$, and that the module of $\times$-indecomposables inherits a $\bullet$-product, and we describe the structure of the resulting ring. In the process, we prove many auxiliary structural results.