Chromatic homotopy theory is asymptotically algebraic

Tobias Barthel; Tomer M. Schlank; Nathaniel Stapleton

doi:10.1007/s00222-019-00943-9

Chromatic homotopy theory is asymptotically algebraic

Tobias Barthel, Tomer M. Schlank, Nathaniel Stapleton

Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

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.

Original language	English
Pages (from-to)	737-845
Number of pages	109
Journal	Inventiones Mathematicae
Volume	220
Issue number	3
DOIs	https://doi.org/10.1007/s00222-019-00943-9
State	Published - Jun 1 2020

Bibliographical note

Funding Information:
Open access funding provided by Projekt DEAL. We would like to thank Mark Behrens, David Gepner, Paul Goerss, Rune Haugseng, Lars Hesselholt, Mike Hopkins, Irakli Patchkoria, and the Homotopy Theory chat room for useful discussions and would all like to thank the MPIM for its hospitality. We are grateful to the referees for many helpful suggestions and corrections. The first author was supported by the DNRF92 and the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant agreement No. 751794. The second author is supported by the Alon fellowship and ISF 1588/18. The third author was supported by SFB 1085 Higher Invariants funded by the DFG and NSF Grant No. DMS-1906236. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Funding Information:
Open access funding provided by Projekt DEAL. We would like to thank Mark Behrens, David Gepner, Paul Goerss, Rune Haugseng, Lars Hesselholt, Mike Hopkins, Irakli Patchkoria, and the Homotopy Theory chat room for useful discussions and would all like to thank the MPIM for its hospitality. We are grateful to the referees for many helpful suggestions and corrections. The first author was supported by the DNRF92 and the European Union?s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant agreement No. 751794. The second author is supported by the Alon fellowship and ISF 1588/18. The third author was supported by SFB 1085 Higher Invariants funded by the DFG and NSF Grant No. DMS-1906236.

Publisher Copyright:
© 2020, The Author(s).

ASJC Scopus subject areas

Mathematics (all)

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10.1007/s00222-019-00943-9

Cite this

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title = "Chromatic homotopy theory is asymptotically algebraic",

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.",

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Chromatic homotopy theory is asymptotically algebraic

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