AbstractThe 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 $$E_\infty $$ E ∞ -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 $$E_\infty $$ E ∞ -ring spectra that exist in the literature are equivalent. We then turn

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