The Balmer spectrum of the equivariant homotopy category of a finite abelian group

TL;DR

This paper determines the structure of the Balmer spectrum for the category of compact genuine A-spectra for finite abelian groups, extending previous results and conjectures, with implications for chromatic fixed-point theory and Tate-constructions.

Contribution

It generalizes the Balmer spectrum description from cyclic groups to all finite abelian groups and proves a corrected log_p-conjecture, advancing the understanding of equivariant stable homotopy theory.

Findings

Complete description of the Balmer spectrum for finite abelian groups.

Proof of the corrected log_p-conjecture for abelian groups.

Generalization of Kuhn's blue-shift theorem for Tate-constructions.

Abstract

For a finite abelian group $A$, we determine the Balmer spectrum of $\mathrm{Sp}_A^{\omega}$, the compact objects in genuine $A$-spectra. This generalizes the case $A=\mathbb{Z}/p\mathbb{Z}$ due to Balmer and Sanders \cite{Balmer-Sanders}, by establishing (a corrected version of) their log$_p$-conjecture for abelian groups. We also work out the consequences for the chromatic type of fixed-points and establish a generalization of Kuhn's blue-shift theorem for Tate-constructions \cite{kuhn}.