The cotangent complex and Thom spectra

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 $G$.

22 pages. Final version