The Stacks project

96.6 The structure sheaf

Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\}$. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. The 2-category of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ has a final object, namely, $\text{id} : (\mathit{Sch}/S)_{fppf} \to (\mathit{Sch}/S)_{fppf}$ and $p$ is a $1$-morphism from $\mathcal{X}$ to this final object. Hence any presheaf $\mathcal{G}$ on $(\mathit{Sch}/S)_{fppf}$ gives a presheaf $p^{-1}\mathcal{G}$ on $\mathcal{X}$ defined by the rule $p^{-1}\mathcal{G}(x) = \mathcal{G}(p(x))$. Moreover, the discussion in Section 96.4 shows that $p^{-1}\mathcal{G}$ is a $\tau$ sheaf whenever $\mathcal{G}$ is a $\tau$-sheaf.

Recall that the site $(\mathit{Sch}/S)_{fppf}$ is a ringed site with structure sheaf $\mathcal{O}$ defined by the rule

see Descent, Definition 35.8.2.

For an object $x$ of $\mathcal{X}$ lying over $U$ we have $\mathcal{O}_\mathcal {X}(x) = \mathcal{O}(U) = \Gamma (U, \mathcal{O}_ U)$. Needless to say $\mathcal{O}_\mathcal {X}$ is also a Zariski, étale, smooth, and syntomic sheaf, and hence each of the sites $\mathcal{X}_{Zar}$, $\mathcal{X}_{\acute{e}tale}$, $\mathcal{X}_{smooth}$, $\mathcal{X}_{syntomic}$, and $\mathcal{X}_{fppf}$ is a ringed site. This construction is functorial as well.