The Stacks project

96.24 Restricting to affines

In this section, given a category $\mathcal{X}$ fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ we will consider the full subcategory $\mathcal{X}_{affine}$ of $\mathcal{X}$ consisting of objects $x$ lying over affine schemes $U$. We will see how, for any topology $\tau$ finer than the Zariski topology, the category of sheaves on $\mathcal{X}$ and $\mathcal{X}_{affine, \tau }$ agree.

Thus the functor $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ restricts to a functor

where the notation on the right hand side is the one introduced in Topologies, Definition 34.7.8. It is clear that $\mathcal{X}_{affine}$ is fibred in groupoids over $(\textit{Aff}/S)_{fppf}$. It follows that $\mathcal{X}_{affine}$ inherits a Zariski, étale, smooth, syntomic, and fppf topology from $(\textit{Aff}/S)_{Zar}$, $(\textit{Aff}/S)_{\acute{e}tale}$, $(\textit{Aff}/S)_{smooth}$, $(\textit{Aff}/S)_{syntomic}$, and $(\textit{Aff}/S)_{fppf}$, see Stacks, Definition 8.10.2.

This definition makes sense by the discussion above. For each $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\}$ a family of morphisms $\{ x_ i \to x\} _{i \in I}$ with fixed target in $\mathcal{X}_{affine}$ is a covering in $\mathcal{X}_{affine, \tau }$ if and only if the family of morphisms $\{ p(x_ i) \to p(x)\} _{i \in I}$ of affine schemes is a standard $\tau$-covering as defined in Topologies, Definitions 34.3.4, 34.4.5, 34.5.5, 34.6.5, and 34.7.5.

Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let us denote $\mathcal{O}$ the restriction of $\mathcal{O}_\mathcal {X}$ to $\mathcal{X}_{affine}$. Then $\mathcal{O}$ is a sheaf in the Zariski, étale, smooth, syntomic, and fppf topologies on $\mathcal{X}_{affine}$. Furthermore, the equivalence of topoi of Lemma 96.24.3 extends to an equivalence

of ringed topoi for $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\}$.