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$E_\infty$-cells and general linear groups of infinite fields

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arxiv 2005.05620 v3 pith:S4ET7EDI submitted 2020-05-12 math.AT math.KT

Soren Galatius , Alexander Kupers , Oscar Randal-Williams This is my paper · ORCID
classification math.ATmath.KT
keywords groupsfieldsgeneralinfiniteinftylinearhomologyline
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abstract

We study the general linear groups of infinite fields (or more generally connected semi-local rings with infinite residue fields) from the perspective of $E_\infty$-algebras. We prove that there is a vanishing line of slope 2 for their $E_\infty$-homology, and analyse the groups on this line by determining all invariant bilinear forms on Steinberg modules. We deduce from this a number of consequences regarding the unstable homology of general linear groups, in particular answering questions of Rognes, Suslin, Mirzaii, and others.

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  1. A stable rank filtration on direct sum $K$-theory

    math.KT 2025-01 conditional novelty 7.0 of 10

    A new stable rank filtration on direct sum K-theory is defined via Gamma-sapces, with filtration quotients described as suspension spectra of decomposition posets.

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