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
is an equivalence.
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