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Journal Article

The cotangent complex and Thom spectra


Stonek,  Bruno
Max Planck Institute for Mathematics, Max Planck Society;

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Rasekh, N., & Stonek, B. (2020). The cotangent complex and Thom spectra. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 90(2), 229-252. doi:10.1007/s12188-020-00226-8.

Cite as: http://hdl.handle.net/21.11116/0000-0007-E535-B
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 $G$.