TY - JOUR

T1 - Chromatic homotopy theory is asymptotically algebraic

AU - Barthel, Tobias

AU - Schlank, Tomer M.

AU - Stapleton, Nathaniel

N1 - Publisher Copyright:
© 2020, The Author(s).

PY - 2020/6/1

Y1 - 2020/6/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85077588573&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85077588573&partnerID=8YFLogxK

U2 - 10.1007/s00222-019-00943-9

DO - 10.1007/s00222-019-00943-9

M3 - Article

AN - SCOPUS:85077588573

SN - 0020-9910

VL - 220

SP - 737

EP - 845

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 3

ER -