Gregory Arone: The Balmer spectrum of functor categories

Abstract.

One of the significant advances in equivariant stable homotopy theory in recent years occurred in the study of the spectrum of tensor triangulated ideals, a.k.a the Balmer spectrum, of compact \(G\)-spectra. This spectrum is now well understood for many groups \(G\), in particular for abelian \(G\).

The category of n-excisive functors from Spectra to Spectra is closed symmetric monoidal under Day convolution. As a stable monoidal category, it has many formal similarities to the category of \(G\)-spectra. For example, compact objects are dualizable. It therefore seems natural to apply the techniques of tensor-triangulated geometry to the study of the category of functors. In this talk we will describe the Balmer spectrum of the category of n-excisive functors. In the process, we describe the analogue of the Burnside ring for excisive functors, which is \(\pi_0\) of the endomorphism ring of the identity. The result also requires calculating the Tate blueshift for the symmetric group with respect to the family of non-transitive subgroups. Joint with Tobias Barthel, Drew Heard and Beren Sanders.