G-typical Witt vectors with coefficients and the norm

2023/05/31 by Thomas T. Read, Read, Thomas · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)

paper · pdf · doi:10.48550/arxiv.2305.20064

openalex publication_date 2023/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a profinite group G we describe an abelian group W_G(R; M) of G-typical Witt vectors with coefficients in an R-module M (where R is a commutative ring). This simultaneously generalises the ring W_G(R) of Dress and Siebeneicher and the Witt vectors with coefficients W(R; M) of Dotto, Krause, Nikolaus and Patchkoria, both of which extend the usual Witt vectors of a ring. We use this new variant of Witt vectors to give a purely algebraic description of the zeroth equivariant stable homotopy groups of the Hill-Hopkins-Ravenel norm N_\e\^G(X) of a connective spectrum X, for any finite group G. Our construction is reasonably analogous to the constructions of previous variants of Witt vectors, and as such is amenable to fairly explicit concrete computations.

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