For a finite abelian group A, we determine the Balmer spectrum of the compact objects in genuine A-spectra. This generalizes the case A=Z/pZ 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)