Skip to Main Content

Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Integral models for spaces via the higher Frobenius
HTML articles powered by AMS MathViewer

by Allen Yuan
J. Amer. Math. Soc. 36 (2023), 107-175
Published electronically: February 14, 2022


We give a fully faithful integral model for simply connected finite complexes in terms of $\mathbb {E}_{\infty }$-ring spectra and the Nikolaus–Scholze Frobenius. The key technical input is the development of a homotopy coherent Frobenius action on a certain subcategory of $p$-complete $\mathbb {E}_{\infty }$-rings for each prime $p$. Using this, we show that the data of a simply connected finite complex $X$ is the data of its Spanier-Whitehead dual, as an $\mathbb {E}_{\infty }$-ring, together with a trivialization of the Frobenius action after completion at each prime.

In producing the above Frobenius action, we explore two ideas which may be of independent interest. The first is a more general action of Frobenius in equivariant homotopy theory; we show that a version of Quillen’s $Q$-construction acts on the $\infty$-category of $\mathbb {E}_{\infty }$-rings with “genuine equivariant multiplication,” which we call global algebras. The second is a “pre-group-completed” variant of algebraic $K$-theory which we call partial $K$-theory. We develop the notion of partial $K$-theory and give a computation of the partial $K$-theory of $\mathbb {F}_p$ up to $p$-completion.

Similar Articles
Bibliographic Information
  • Allen Yuan
  • Affiliation: Department of Mathematics, Columbia University. 2990 Broadway, New York, New York 10027
  • MR Author ID: 964228
  • Received by editor(s): October 9, 2019
  • Received by editor(s) in revised form: September 23, 2021
  • Published electronically: February 14, 2022
  • Additional Notes: The author was supported by the NSF under Grant DGE-1122374
  • © Copyright 2022 by the author
  • Journal: J. Amer. Math. Soc. 36 (2023), 107-175
  • MSC (2020): Primary 55P43, 55P15; Secondary 55P91, 19D06
  • DOI:
  • MathSciNet review: 4495840