Tomer Schlank: Ultra-Products and Chromatic Homotopy Theory

Let \(C_{p,n}\)be the \(K(n)\)-local category at height \(n\) and prime \(p\). These categories are of great interest to the stable homotopy theorist since they serve as a the “associated  graded” pieces of the chromatic filtration on the category of spectra. It is a well known observation that for a given height \(n\) certain “special” phenomena happen only for small enough primes. Further, in some sense, the categories \(C_{p,n}\) become more regular and algebraic as \(p\) goes to infinity for a fixed \(n\). The goal of this talk is to make this intuition precise.

Given an infinite sequence of mathematical structures, logicians have a method to construct a limiting one by using “ultra-products”. We shall define a notion of “ultra-product of categories” and then describe a collection of categories \(D_{n,p}\) which will serve as algebro-geometric analogs of the \(K(n)\)-local category at the prime \(p\).

Then for a  fixed height \(n\) we prove:

\(\prod_p^{\mathrm{Ultra}} C_{n,p} \cong \prod_p^{\mathrm{Ultra}} D_{n,p}.\)

If time permits we shall describe our ongoing attempts to use these methods to get a version of the \(K(n)\)-local category corresponding to formal Drinfeld modules (instead of formal groups).

This is a joint project with N. Stapleton and T. Barthel.