A short proof of the chromatic Smith Fixed Point Theorem

Abstract

We give a short and much simplified proof of the main theorem of the recent study, by T. Barthel, M. Hausmann, N. Naumann, T. Nikolaus, J. Noel, and N. Stapleton, of the Balmer spectrum for A-equivariant stable homotopy when A is a finite abelian p-group. This theorem says that if A is a finite abelian p-group of rank r, and X is a finite A-space that is acyclic in the (n+r)th Morava K-theory, then its space of fixed points, X^A, will be acyclic in the nth Morava K-theory. It is a chromatic homotopy version of P. A. Smith's classic theorem about the mod p homology of the fixed points of a finite A-space.