概览概形的切范畴
A Deep Dive Into the Tangent Category of Schemes
AI总结：
本文深入探究概形范畴的切结构，明确了两类双纤维化的构建方式，给出切结构的描述，并证明拟分离概形可由其微分层范畴唯一确定。
AI中文摘要：
本文是一篇说明性论文，深入且明确地探究并阐述了概形范畴$\boldsymbol{Sch}_{/S}$上的切结构，其中切函子$T(X)=T_{X/S}$是格罗滕迪克在《代数几何基础》第4卷中定义的相对切概形。特别地，本文明确阐述了概形上拟凝聚层的双纤维化与拟凝聚代数层的双纤维化，如何由交换环上模的双纤维化与交换代数的双纤维化的相互作用构建而成。本文还展示了这些相互作用如何通过凯勒微分层、相对谱函子的性质等，给出概形范畴上标准切结构的明确描述。最后，本文证明对于拟分离概形$X$和$Y$，存在范畴等价$\boldsymbol{DBun}(X)\backsimeq\boldsymbol{DBun}(Y)$当且仅当存在概形同构$X\backsimeq Y$，即拟凝聚层可由其微分层范畴重构。
英文摘要：
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{Éléments de Géométrie Algébrique} 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 Kähler 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 $\mathbf{DBun}(X) \simeq \mathbf{DBun}(Y)$.
