A Deep Dive Into the Tangent Category of Schemes

Abstract

In this largely expository paper we provide a deep and explicit exploration and exposition of the tangent structure on the category of schemes $\mathbf{Sch}_{/S}$ whose tangent functor $T(X) = T_{X/S}$ is the relative tangent scheme of Grothendieck described in \emph{\'El\'ements de G\'eom\'etrie Alg\'ebrique} 4. In particular we provide explicit descriptions of the ways that the bifibration of quasicoherent sheaves and bifbration of quesicoherent sheaves of algebras over schemes may be built from the ways in which the bifibrations of modules and commutative algebras over commutative rings interact. We also show the ways in which these interactions give rise to an explicit description of the standard tangent structure on the category of schemes in terms of sheaves of K\"ahler differentials, properties of the relative spectrum functor, and more. Finally, we show that quasi-coherent sheaves can be reconstructed from their category of differential bundles by showing that for quasi-separated schemes $X$ and $Y$, there is an isomorphism $X \cong Y$ if and only if there is an equivalence of categories $\mathbf{DBun}(X) \simeq \mathbf{DBun}(Y)$.