Download this article
Download this article For screen
For printing
Recent Issues

Volume 23
Issue 9, 3909–4400
Issue 8, 3417–3908
Issue 7, 2925–3415
Issue 6, 2415–2924
Issue 5, 1935–2414
Issue 4, 1463–1934
Issue 3, 963–1462
Issue 2, 509–962
Issue 1, 1–508

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
Other MSP Journals
This article is available for purchase or by subscription. See below.
Thom spectra, higher $\mathrm{THH}$ and tensors in $\infty$–categories

Nima Rasekh, Bruno Stonek and Gabriel Valenzuela

Algebraic & Geometric Topology 22 (2022) 1841–1903

Let f : G Pic (R) be a map of E–groups, where Pic (R) denotes the Picard space of an E–ring spectrum R. We determine the tensor X RMf of the Thom ER–algebra Mf with a space X; when X is the circle, the tensor with X is topological Hochschild homology over R. We use the theory of localizations of –categories as a technical tool: we contribute to this theory an –categorical analogue of Day’s reflection theorem about closed symmetric monoidal structures on localizations, and we prove that, for a smashing localization L of the –category of presentable –categories, the free L–local presentable –category on a small simplicial set K is given by presheaves on K valued on the L–localization of the –category of spaces.

If X is a pointed space, a map g: A B of E–ring spectra satisfies X–base change if X B is the pushout of A X A along g. Building on a result of Mathew, we prove that if g is étale then it satisfies X–base change provided X is connected. We also prove that g satisfies X–base change provided the multiplication map of B is an equivalence. Finally, we prove that, under some hypotheses, the Thom isomorphism of Mahowald cannot be an instance of S0–base change.

PDF Access Denied

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Thom spectra, topological Hochschild homology, $E_\infty$ ring spectra, Thom isomorphism, topological K–theory
Mathematical Subject Classification
Primary: 55P43
Secondary: 18D20, 55N20
Received: 8 April 2020
Revised: 29 December 2020
Accepted: 29 March 2021
Published: 10 October 2022
Nima Rasekh
Max Planck Institute for Mathematics
Bruno Stonek
University of Warsaw
Gabriel Valenzuela
Max Planck Institute for Mathematics