Maps between spherical group rings

TL;DR

This paper characterizes the space of $ ext{E}_ infty$-ring maps between spherical group rings of finitely generated abelian groups, showing it corresponds exactly to group homomorphisms, and extends results to other ring spectra.

Contribution

It establishes a precise correspondence between $ ext{E}_ infty$-ring maps and group homomorphisms for spherical group rings, and generalizes to other spectra like $p$-groups and chromatically complete rings.

Findings

The space of $ ext{E}_ infty$-ring maps between $ ext{S}[A]$ and $ ext{S}[B]$ is discrete and corresponds to group homomorphisms.

Provides a formula for strict units in group rings of finite $p$-groups over $p$-complete spectra.

Extends the understanding of maps between ring spectra beyond classical cases.

Abstract

We prove that for finitely generated abelian groups $A$ and $B$, the space of $\mathbb{E}_\infty$-ring maps between the spherical groups rings $\mathbb{S}[A] \to \mathbb{S}[B]$ is equivalent to the discrete set of group homomorphisms $A \to B$. We also prove generalizations where the sphere is replaced by other ring spectra, e.g. we give a formula for the strict units in group rings of the form $R[A]$ for $A$ a finite $p$-group and $R$ $p$-completely chromatically complete.