The cotangent complex and Thom spectra
| Autor: | Bruno Stonek, Nima Rasekh |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Calculus of functors General Mathematics Context (language use) Commutative ring 01 natural sciences Spectrum (topology) 55P43 (Primary) 14F10 (Secondary) Mathematics - Algebraic Geometry higher category theory 0103 physical sciences FOS: Mathematics Algebraic Topology (math.AT) Cotangent complex Mathematics - Algebraic Topology 0101 mathematics Algebraic Geometry (math.AG) Mathematics Ring (mathematics) Smash product 010102 general mathematics thom spectra Cobordism structured ring spectra goodwillie calculus cotangent complex 010307 mathematical physics |
| Zdroj: | Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg |
| ISSN: | 1865-8784 0025-5858 |
| DOI: | 10.1007/s12188-020-00226-8 |
| Popis: | 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 $G$. Comment: 22 pages. Final version |
| Databáze: | OpenAIRE |
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