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 $\mathcal{O}_\mathcal {X} = p^{-1}\mathcal{O}$.

## 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

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.

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.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)