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Chromatic (co)homology of finite general linear groups

### Samuel M A Hutchinson, Samuel J Marsh and Neil P Strickland

Algebraic & Geometric Topology 22 (2022) 1511–1614
##### Abstract

We study the Morava $E$–theory (at a prime $p$) of $B{\mathrm{GL}}_{d}\left(F\right)$, where $F$ is a finite field with $|F|=1\phantom{\rule{0.3em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.3em}{0ex}}p\right)$. Taking all $d$ together, we obtain a structure with two products $×$ and $\bullet$. We prove that it is a polynomial ring under $×$ and that the module of $×$–indecomposables inherits a $\bullet$–product, and we describe the structure of the resulting ring. In the process, we prove many auxiliary structural results.

##### Keywords
Morava K–theory, general linear groups
##### Mathematical Subject Classification 2010
Primary: 55N20
Secondary: 14L05, 55N22, 55R35