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