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

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}.

v3: minor changes, to appear in Inventiones Mathematicae