Thom spectra, higher THH and tensors in ∞–categories

Rasekh, Nima; Stonek, Bruno; Valenzuela, Gabriel

doi:10.2140/agt.2022.22.1841

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Thom spectra, higher $\mathrm{THH}$ and tensors in $\infty$–categories

Nima Rasekh, Bruno Stonek and Gabriel Valenzuela

Algebraic & Geometric Topology 22 (2022) 1841–1903

DOI: 10.2140/agt.2022.22.1841

Abstract

Let $f : G \to Pic (R)$ be a map of $E_{\infty}$ –groups, where $Pic (R)$ denotes the Picard space of an $E_{\infty}$ –ring spectrum $R$ . We determine the tensor $X \otimes_{R} M f$ of the Thom $E_{\infty}$ – $R$ –algebra $M f$ 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 $\infty$ –categories as a technical tool: we contribute to this theory an $\infty$ –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 $\infty$ –category of presentable $\infty$ –categories, the free $L$ –local presentable $\infty$ –category on a small simplicial set $K$ is given by presheaves on $K$ valued on the $L$ –localization of the $\infty$ –category of spaces.

If $X$ is a pointed space, a map $g : A \to B$ of $E_{\infty}$ –ring spectra satisfies $X$ –base change if $X \otimes B$ is the pushout of $A \to X \otimes A$ 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 $S^{0}$ –base change.

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Keywords

Thom spectra, topological Hochschild homology, $E_\infty$ ring spectra, Thom isomorphism, topological K–theory

Mathematical Subject Classification

Primary: 55P43

Secondary: 18D20, 55N20

References

Publication

Received: 8 April 2020

Revised: 29 December 2020

Accepted: 29 March 2021

Published: 10 October 2022

Authors

Nima Rasekh
Max Planck Institute for Mathematics Bonn Germany

Bruno Stonek
University of Warsaw Warszawa Poland

Gabriel Valenzuela
Max Planck Institute for Mathematics Bonn Germany

Volume 22, issue 4 (2022)