Chromatic homotopy theory is asymptotically algebraic

• 成果类型: Article
• 署名作者: Barthel, Tobias; Schlank, Tomer; Stapleton, Nathaniel
• 署名单位: Max Planck Society; Hebrew University of Jerusalem; University of Kentucky
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00943-9
• 发表日期: 2020
• 页码: 737-845
• 关键词: k(2)-local sphere; CATEGORIES; HOMOLOGY; localization; SPECTRA; MODULES; MODEL

摘要

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.