The Stacks project

Lemma 18.23.4. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal {D}$-module.

  1. If $\mathcal{F}$ is locally free then $f^*\mathcal{F}$ is locally free.

  2. If $\mathcal{F}$ is finite locally free then $f^*\mathcal{F}$ is finite locally free.

  3. If $\mathcal{F}$ is locally generated by sections then $f^*\mathcal{F}$ is locally generated by sections.

  4. If $\mathcal{F}$ is locally generated by $r$ sections then $f^*\mathcal{F}$ is locally generated by $r$ sections.

  5. If $\mathcal{F}$ is of finite type then $f^*\mathcal{F}$ is of finite type.

  6. If $\mathcal{F}$ is quasi-coherent then $f^*\mathcal{F}$ is quasi-coherent.

  7. If $\mathcal{F}$ is of finite presentation then $f^*\mathcal{F}$ is of finite presentation.

Proof. According to the discussion in Section 18.18 we need only check preservation under pullback for a morphism of ringed sites $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ such that $f$ is given by a left exact, continuous functor $u : \mathcal{D} \to \mathcal{C}$ between sites which have all finite limits. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_\mathcal {D}$-modules which has one of the properties (1) – (6) of Definition 18.23.1. We know $\mathcal{D}$ has a final object $Y$ and $X = u(Y)$ is a final object for $\mathcal{C}$. By assumption we have a covering $\{ Y_ i \to Y\} $ such that $\mathcal{G}|_{\mathcal{D}/Y_ i}$ has the corresponding global property. Set $X_ i = u(Y_ i)$ so that $\{ X_ i \to X\} $ is a covering in $\mathcal{C}$. We get a commutative diagram of morphisms ringed sites

\[ \xymatrix{ (\mathcal{C}/X_ i, \mathcal{O}_\mathcal {C}|_{X_ i}) \ar[r] \ar[d] & (\mathcal{C}, \mathcal{O}_\mathcal {C}) \ar[d] \\ (\mathcal{D}/Y_ i, \mathcal{O}_\mathcal {D}|_{Y_ i}) \ar[r] & (\mathcal{D}, \mathcal{O}_\mathcal {D}) } \]

by Sites, Lemma 7.28.2. Hence by Lemma 18.17.2 that $f^*\mathcal{G}|_{X_ i}$ has the corresponding global property. Hence we conclude that $\mathcal{G}$ has the local property we started out with by Lemma 18.23.3. $\square$

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