Section 46.2 (06Z3): Conventions

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46.2 Conventions

In this chapter we fix $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\}$ and we fix a big $\tau$ -site $\mathit{Sch}_\tau$ as in Topologies, Section 34.2. All schemes will be objects of $\mathit{Sch}_\tau$ . In particular, given a scheme $S$ we obtain sites $(\textit{Aff}/S)_\tau \subset (\mathit{Sch}/S)_\tau$ . The structure sheaf $\mathcal{O}$ on these sites is defined by the rule $\mathcal{O}(T) = \Gamma (T, \mathcal{O}_ T)$ .

All rings $A$ will be such that $\mathop{\mathrm{Spec}}(A)$ is (isomorphic to) an object of $\mathit{Sch}_\tau$ . Given a ring $A$ we denote $\textit{Alg}_ A$ the category of $A$ -algebras whose objects are the $A$ -algebras $B$ of the form $B = \Gamma (U, \mathcal{O}_ U)$ where $S$ is an affine object of $\mathit{Sch}_\tau$ . Thus given an affine scheme $S = \mathop{\mathrm{Spec}}(A)$ the functor

$(\textit{Aff}/S)_\tau \longrightarrow \textit{Alg}_ A, \quad U \longmapsto \mathcal{O}(U)$

is an equivalence.

Comments (1)

Comment #9947 by Luming Zhao on January 24, 2025 at 08:33

In the last but one sentence "...where $S$ is an affine object of $Sch_{\tau}$ .", one should replace $S$ by $U$ ?

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46.2 Conventions

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