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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
(Invent Math 208(1):283-326, 2017), 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
(Kuhn in Invent Math 157(2):345–370,
2004).