The Stacks project

94.6 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 94.4 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 94.6.1. 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 94.6.2. 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 $f^{-1}\mathcal{O}_\mathcal {Y} = \mathcal{O}_\mathcal {X}$ 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 $p = q \circ f$, hence $p^{-1} = f^{-1} \circ q^{-1}$ by Lemma 94.3.2. Since $\mathcal{O}_\mathcal {X} = p^{-1}\mathcal{O}$ and $\mathcal{O}_\mathcal {Y} = q^{-1}\mathcal{O}$ the result follows. $\square$

Remark 94.6.3. In the situation of Lemma 94.6.2 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.

Comments (2)

Comment #6537 by Hadi Hedayatzadeh on

In Lemma 06TW, the canonical identification should be (for some reason it does render the math equation...)

Also, it the proof of the lemma, the composition should be

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