Eilenberg Mac Lane spectra as p-cyclonic Thom spectra

Hopkins and Mahowald gave a simple description of the mod \(p\) Eilenberg Mac Lane spectrum \(\mathbb{F}_p\) as the free \(\mathbb{E}_2\)-algebra with an equivalence of \(p\) and \(0\). We show for each faithful \(2\)-dimensional representation \(\lambda\) of a \(p\)-group \(G\) that the \(G\)-equiv...

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Bibliographic Details
Published in:arXiv.org 2021-10
Main Author: Levy, Ishan
Format: Article
Language:English
Subjects:
Algebra
Equivalence
Representations
Subgroups
Online Access:Get full text
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Summary:Hopkins and Mahowald gave a simple description of the mod \(p\) Eilenberg Mac Lane spectrum \(\mathbb{F}_p\) as the free \(\mathbb{E}_2\)-algebra with an equivalence of \(p\) and \(0\). We show for each faithful \(2\)-dimensional representation \(\lambda\) of a \(p\)-group \(G\) that the \(G\)-equivariant Eilenberg Mac Lane spectrum \(\underline{\mathbb{F}}_p\) is the free \(\mathbb{E}_{\lambda}\)-algebra with an equivalence of \(p\) and \(0\). This unifies and simplifies recent work of Behrens, Hahn, and Wilson, and extends it to include the dihedral \(2\)-subgroups of O(2). The main new idea is that \(\underline{\mathbb{F}}_p\) has a simple description as a \(p\)-cyclonic module over \(\rm{THH}(\mathbb{F}_p)\). We show our result is the best possible one in that it gives all groups \(G\) and representations \(V\) such that \(\underline{\mathbb{F}}_p\) is the free \(\mathbb{E}_V\)-algebra with an equivalence of \(p\) and \(0\).
ISSN:2331-8422