A short proof of the chromatic Smith Fixed Point Theorem

TL;DR

This paper presents a concise and simplified proof of a key theorem in chromatic homotopy theory, relating the Morava K-theory acyclicity of finite A-spaces to their fixed points for finite abelian p-groups.

Contribution

It provides a shorter, more accessible proof of the chromatic Smith Fixed Point Theorem for finite abelian p-groups, advancing understanding in equivariant stable homotopy theory.

Findings

Fixed points of A-spaces are acyclic in lower Morava K-theory levels.

The proof simplifies the understanding of the Balmer spectrum in equivariant homotopy.

The theorem generalizes P. A. Smith's classical fixed point homology result.

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.