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The cotangent complex and Thom spectra

Rasekh, N., & Stonek, B. (in press). The cotangent complex and Thom spectra. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, Published Online - Print pending. doi:10.1007/s12188-020-00226-8.

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Genre: Journal Article

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Locator:
https://doi.org/10.1007/s12188-020-00226-8 (Publisher version)
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Creators:
Rasekh, Nima, Author
Stonek, Bruno1, Author
Affiliations:
1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201

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Free keywords: Mathematics, Algebraic Topology, Algebraic Geometry
Abstract: The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of $E_\infty$-ring spectra in various ways. In this work we first establish, in the context of $\infty$-categories and using Goodwillie's calculus of functors, that various definitions of the cotangent complex of a map of $E_\infty$-ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let $R$ be an $E_\infty$-ring spectrum and $\mathrm{Pic}(R)$ denote its Picard $E_\infty$-group. Let $Mf$ denote the Thom $E_\infty$-$R$-algebra of a map of $E_\infty$-groups $f:G\to \mathrm{Pic}(R)$; examples of $Mf$ are given by various flavors of cobordism spectra. We prove that the cotangent complex of $R\to Mf$ is equivalent to the smash product of $Mf$ and the connective spectrum associated to $G$.

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Language(s): eng - English
Dates: 2020
Publication Status: Accepted / In Press
Pages: 24
Publishing info: -
Rev. Type: Peer
Identifiers: arXiv: 2005.01382
DOI: 10.1007/s12188-020-00226-8
Degree: -

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Title: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
Source Genre: Journal
Creator(s):
Affiliations:
Publ. Info: Springer
Pages: - Volume / Issue: - Sequence Number: Published Online - Print pending Start / End Page: - Identifier: -