Definition 96.6.1. 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 $\mathcal{O}_\mathcal {X} = p^{-1}\mathcal{O}$.
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
\[ (\mathit{Sch}/S)^{opp} \longrightarrow \textit{Rings}, \quad U/S \longmapsto \Gamma (U, \mathcal{O}_ U) \]
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 96.6.2. 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 96.3.2. Since $\mathcal{O}_\mathcal {X} = p^{-1}\mathcal{O}$ and $\mathcal{O}_\mathcal {Y} = q^{-1}\mathcal{O}$ the result follows. $\square$
Remark 96.6.3. In the situation of Lemma 96.6.2 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).
All contributions are licensed under the GNU Free Documentation License.
Comments (2)
Comment #6537 by Hadi Hedayatzadeh on
Comment #6589 by Johan on