Ambidextrous global spectra and tempered cohomology

TL;DR

This paper introduces a new framework for global equivariant spectra with additional transfer structures, applies it to tempered cohomology theories related to P-divisible groups, and explores their properties and fixed points.

Contribution

It develops the concept of $Q$-ambidextrous global spectra, constructs representations for tempered cohomology theories, and analyzes their fixed point properties in equivariant stable homotopy theory.

Findings

Tempered cohomology theories are represented by $pi$-ambidextrous global $E_$ ring spectra.

Fixed points vanish for nonabelian groups and have a simple model for abelian groups.

Constructed a well-behaved $F$-global homology theory with base change properties.

Abstract

We introduce generalizations of global equivariant spectra which encode globally equivariant cohomology theories equipped with additional transfers, such as the deflation maps present in equivariant topological $K$-theory. We call these $\mathcal{Q}$-ambidextrous global spectra, where $\mathcal{Q}$ is a parameter encoding which additional transfers one allows. As our main example, we prove that the tempered cohomology theory associated with an oriented $\mathbf{P}$-divisible group, constructed by Lurie, is represented by a $\pi$-ambidextrous global $\mathbf{E}_\infty$ ring spectrum, encoding transfers along all relatively $\pi$-finite maps of global spaces. This is established by means of a general parametrized decategorification process, perhaps of independent interest, that produces $\mathcal{Q}$-ambidextrous global spectra from suitable global families of stable $\infty$-categories. By allowing $\mathcal{Q}$ to vary, we are able to coherently encode the fact that non-invertible morphisms of oriented $\mathbf{P}$-divisible groups induce maps of tempered theories that only commute with certain transfers. With these $\pi$-ambidextrous enhancements in hand, we explore the fundamental properties of tempered theories as equivariant stable homotopy types. We construct a well-behaved $F$-global homology theory for any $\pi$-finite space $F$, with good base change properties. Taking $F = \mathbf{B} H$ for a finite group $H$, this establishes general base change results for the geometric fixed points of tempered theories. We use this to compute the $H$-geometric fixed points of tempered theories, showing that they vanish for $H$ nonabelian and admit a simple algebro-geometric model when $H$ is abelian, with identifiable blueshift properties.