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 \(f:G\rightarrow \mathrm {Pic}(R)\); examples of Mf are given by various flavors of cobordism spectra. We prove that the cotangent complex of \(R\rightarrow Mf\) is equivalent to the smash product of Mf and the connective spectrum associated to G.