The Stacks project

93.6 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 93.4 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

\[ (\mathit{Sch}/S)^{opp} \longrightarrow \textit{Rings}, \quad U/S \longmapsto \Gamma (U, \mathcal{O}_ U) \]

see Descent, Definition 35.8.2.

Definition 93.6.1. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. The structure sheaf of $\mathcal{X}$ is the sheaf of rings $\mathcal{O}_\mathcal {X} = p^{-1}\mathcal{O}$.

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, ├ętale, 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.

Lemma 93.6.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $. There is a canonical identification $f^{-1}\mathcal{O}_\mathcal {X} = \mathcal{O}_\mathcal {Y}$ which turns $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_\tau ) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_\tau )$ into a morphism of ringed topoi.

Proof. Denote $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ and $q : \mathcal{Y} \to (\mathit{Sch}/S)_{fppf}$ the structural functors. Then $q = p \circ f$, hence $q^{-1} = f^{-1} \circ p^{-1}$ by Lemma 93.3.2. The result follows. $\square$

Remark 93.6.3. In the situation of Lemma 93.6.2 the morphism of ringed topoi $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_\tau ) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{Y}_\tau )$ is flat as is clear from the equality $f^{-1}\mathcal{O}_\mathcal {X} = \mathcal{O}_\mathcal {Y}$. This is a bit counter intuitive, for example because a closed immersion of algebraic stacks is typically not flat (as a morphism of algebraic stacks). However, exactly the same thing happens when taking a closed immersion $i : X \to Y$ of schemes: in this case the associated morphism of big $\tau $-sites $i : (\mathit{Sch}/X)_\tau \to (\mathit{Sch}/Y)_\tau $ also is flat.


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