Abstract
Inspired by the Ax–Kochen isomorphism theorem, we develop a notion of categorical ultraproducts to capture the generic behavior of an infinite collection of mathematical objects. We employ this theory to give an asymptotic solution to the approximation problem in chromatic homotopy theory. More precisely, we show that the ultraproduct of the E(n, p)-local categories over any non-principal ultrafilter on the set of prime numbers is equivalent to the ultraproduct of certain algebraic categories introduced by Franke. This shows that chromatic homotopy theory at a fixed height is asymptotically algebraic.
Highlights
It is possible to algebraically model partial information about Spn,p
To describe the algebraic approximations we will use, recall that Quillen’s work on complex cobordism reveals a close connection between stable homotopy theory and the moduli stack of formal groups Mfg: the cohomology of the tensor powers of the canonical line bundle on the moduli stack forms the E2-page of the Adams–Novikov spectral sequence converging to π∗S0
The p-local moduli stack (Mfg)p admits an increasing filtration by the open substacks (Mfg)n,p consisting of formal groups of height ≤ n at the prime p that mirrors the chromatic filtration as observed by Morava
Summary
In this subsection we explain the basics of ultrafilters and ultraproducts. Our goal is to give the background necessary for the paper and a brief introduction for the working homotopy theorist. Lemma 2.2 If F is a filter on I , there exists an ultrafilter F on I containing F. Lemma 2.5 An ultrafilter F that contains a finite set is principal. Lemma 2.6 An ultrafilter F is non-principal if and only if it contains F∞ Proof This follows immediately from Lemma 2.5. Lemma 2.7 If A ⊆ I is infinite, there exists a non-principal ultrafilter F on I such that A ∈ F. Proof Consider the collection of subsets of I that contain all but a finite number of elements in A. This is a filter that contains F∞ and by Lemma 2.2 it can be completed to an ultrafilter
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