We first prove, 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 prove the following theorem: if $R$ is an $E_\infty$-ring spectrum and
$f:G\to \mathrm{Pic}(R)$ is a map of $E_\infty$-groups, then the cotangent
complex over $R$ of the Thom $E_\infty$-$R$-algebra of $f$ is equivalent to the
smash product of $Mf$ and the connective spectrum associated to $G$.Comment: 19 page

Rasekh, Nima

Stonek, Bruno

English

arXiv

The cotangent complex of a map of commutative rings is a central object indeformation theory. Since the 1990s, it has been generalized to the homotopicalsetting 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 theliterature 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 connectivespectrum associated to $G$.<br

Rasekh, N.

Stonek, B. 

MPG.PuRe

The cotangent complex and Thom spectra

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-infinity-ring spectra in various ways. In this work we first establish, in the context of infinity-categories and using Goodwillie's calculus of functors, that various definitions of the cotangent complex of a map of E-infinity-ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let R be an E-infinity-ring spectrum and Pic(R) denote its Picard E8-group. Let M f denote the Thom E-infinity- R-algebra of a map of E-infinity-groups f : G. Pic(R); examples of M f are given by various flavors of cobordism spectra. We prove that the cotangent complex of R -> M f is equivalent to the smash product of M f and the connective spectrum associated to G.UPHES

Infoscience - École polytechnique fédérale de Lausanne

http://hdl.handle.net/21.11116/0000-0008-2861-E

The cotangent complex and Thom spectra

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MPG.PuRe

Infoscience - École polytechnique fédérale de Lausanne