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.