The cotangent complex and Thom spectra [article]

Nima Rasekh, Bruno Stonek
2020 arXiv   pre-print
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
more » ... ing 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.
arXiv:2005.01382v2 fatcat:pcuelhxdwfd27k2e7pp43udtri