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.

交换环图的余切复数是变形理论的中心对象。自1990年代以来，它已以各种方式推广到\（E_ \ infty \） -环谱的同位设置。在这项工作中，我们首先在\（\ infty \）-类别的背景下，并使用Goodwillie的函子演算，建立了\（E_ \ infty \） -环谱图的余切复合物的各种定义。文献是等效的。然后，我们将注意力转向一个特定的示例。令R为\（E_ \ infty \） -环谱，\（\ mathrm {Pic}（R）\）表示其Picard \（E_ \ infty \）-组。让Mf表示汤姆\（E_ \ infty \） - [R -代数的地图上\（E_ \ infty \） -基团\（F：G \ RIGHTARROW \ mathrm {产品图}（R）\） ; Mf的例子由不同种类的Cobordism光谱给出。我们证明\（R \ rightarrow Mf \）的余切复数等于Mf的粉碎乘积以及与G相关的连接谱。