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_∞ 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_∞ E ∞ -ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let R be an E_∞ E ∞ -ring spectrum and Pic(R) Pic ( R ) denote its Picard E_∞ E ∞ -group. Let Mf denote the Thom E_∞ E ∞ - R -algebra of a map of E_∞ E ∞ -groups f:G→ Pic(R) f : G → Pic ( R ) ; examples of Mf are given by various flavors of cobordism spectra. We prove that the cotangent complex of R→ Mf R → M f is equivalent to the smash product of Mf and the connective spectrum associated to G .