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Mathematics, Algebraic Topology, Algebraic Geometry
Abstract:
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 $f:G\to \mathrm{Pic}(R)$; examples of $Mf$ are given by
various flavors of cobordism spectra. We prove that the cotangent complex of
$R\to Mf$ is equivalent to the smash product of $Mf$ and the connective
spectrum associated to $G$.
