The Stacks project

Proof. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a special cocontinuous functor. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules on $\mathcal{C}$. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the equivalence of topoi associated to $u$. Set $\mathcal{O}' = g_*\mathcal{O}$, and let $g^\sharp : \mathcal{O}' \to g_*\mathcal{O}$ be the identity. Finally, set $\mathcal{F}' = g_*\mathcal{F}$. Let $\mathcal{P}_ l$ be one of the properties (1) – (7) listed in Definition 18.23.1. (We will discuss the coherent case at the end of the proof.) Let $\mathcal{P}_ g$ denote the corresponding property listed in Definition 18.17.1. We have already seen that $\mathcal{P}_ g$ is intrinsic. We have to show that $\mathcal{P}_ l(\mathcal{C}, \mathcal{O}, \mathcal{F})$ holds if and only if $\mathcal{P}_ l(\mathcal{D}, \mathcal{O}', \mathcal{F}')$ holds.

Assume that $\mathcal{F}$ has $\mathcal{P}_ l$. Let $V$ be an object of $\mathcal{D}$. One of the properties of a special cocontinuous functor is that there exists a covering $\{ u(U_ i) \to V\} _{i \in I}$ in the site $\mathcal{D}$. By assumption, for each $i$ there exists a covering $\{ U_{ij} \to U_ i\} _{j \in J_ i}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{U_{ij}}$ is $\mathcal{P}_ g$. By Sites, Lemma 7.29.3 we have commutative diagrams of ringed topoi

where the vertical arrows are equivalences. Hence we conclude that $\mathcal{F}'|_{u(U_{ij})}$ has property $\mathcal{P}_ g$ also. And moreover, $\{ u(U_{ij}) \to V\} _{i \in I, j \in J_ i}$ is a covering of the site $\mathcal{D}$. Hence $\mathcal{F}'$ has property $\mathcal{P}_ l$.

Assume that $\mathcal{F}'$ has $\mathcal{P}_ l$. Let $U$ be an object of $\mathcal{C}$. By assumption, there exists a covering $\{ V_ i \to u(U)\} _{i \in I}$ such that $\mathcal{F}'|_{V_ i}$ has property $\mathcal{P}_ g$. Because $u$ is cocontinuous we can refine this covering by a family $\{ u(U_ j) \to u(U)\} _{j \in J}$ where $\{ U_ j \to U\} _{j \in J}$ is a covering in $\mathcal{C}$. Say the refinement is given by $\alpha : J \to I$ and $u(U_ j) \to V_{\alpha (j)}$. Restricting is transitive, i.e., $(\mathcal{F}'|_{V_{\alpha (j)}})|_{u(U_ j)} = \mathcal{F}'|_{u(U_ j)}$. Hence by Lemma 18.17.2 we see that $\mathcal{F}'|_{u(U_ j)}$ has property $\mathcal{P}_ g$. Hence the diagram

where the vertical arrows are equivalences shows that $\mathcal{F}|_{U_ j}$ has property $\mathcal{P}_ g$ also. Thus $\mathcal{F}$ has property $\mathcal{P}_ l$ as desired.

Finally, we prove the lemma in case $\mathcal{P}_ l = coherent$¹. Assume $\mathcal{F}$ is coherent. This implies that $\mathcal{F}$ is of finite type and hence $\mathcal{F}'$ is of finite type also by the first part of the proof. Let $V$ be an object of $\mathcal{D}$ and let $s_1, \ldots , s_ n \in \mathcal{F}'(V)$. We have to show that the kernel $\mathcal{K}'$ of $\bigoplus _{j = 1, \ldots , n} \mathcal{O}_ V \to \mathcal{F}'|_ V$ is of finite type on $\mathcal{D}/V$. This means we have to show that for any $V'/V$ there exists a covering $\{ V'_ i \to V'\} $ such that $\mathcal{F}'|_{V'_ i}$ is generated by finitely many sections. Replacing $V$ by $V'$ (and restricting the sections $s_ j$ to $V'$) we reduce to the case where $V' = V$. Since $u$ is a special cocontinuous functor, there exists a covering $\{ u(U_ i) \to V\} _{i \in I}$ in the site $\mathcal{D}$. Using the isomorphism of topoi $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i) = \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/u(U_ i))$ we see that $\mathcal{K}'|_{u(U_ i)}$ corresponds to the kernel $\mathcal{K}_ i$ of a map $\bigoplus _{j = 1, \ldots , n} \mathcal{O}_{U_ i} \to \mathcal{F}|_{U_ i}$. Since $\mathcal{F}$ is coherent we see that $\mathcal{K}_ i$ is of finite type. Hence we conclude (by the first part of the proof again) that $\mathcal{K}|_{u(U_ i)}$ is of finite type. Thus there exist coverings $\{ V_{il} \to u(U_ i)\} $ such that $\mathcal{K}|_{V_{il}}$ is generated by finitely many global sections. Since $\{ V_{il} \to V\} $ is a covering of $\mathcal{D}$ we conclude that $\mathcal{K}$ is of finite type as desired.

Assume $\mathcal{F}'$ is coherent. This implies that $\mathcal{F}'$ is of finite type and hence $\mathcal{F}$ is of finite type also by the first part of the proof. Let $U$ be an object of $\mathcal{C}$, and let $s_1, \ldots , s_ n \in \mathcal{F}(U)$. We have to show that the kernel $\mathcal{K}$ of $\bigoplus _{j = 1, \ldots , n} \mathcal{O}_ U \to \mathcal{F}|_ U$ is of finite type on $\mathcal{C}/U$. Using the isomorphism of topoi $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) = \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/u(U))$ we see that $\mathcal{K}|_ U$ corresponds to the kernel $\mathcal{K}'$ of a map $\bigoplus _{j = 1, \ldots , n} \mathcal{O}_{u(U)} \to \mathcal{F}'|_{u(U)}$. As $\mathcal{F}'$ is coherent, we see that $\mathcal{K}'$ is of finite type. Hence, by the first part of the proof again, we conclude that $\mathcal{K}$ is of finite type. $\square$

Proof. In each case we have (a) $\Rightarrow (b)$. In each of the cases (1) - (6) condition (b) implies condition (c) by axiom (2) of a site (see Sites, Definition 7.6.2) and the definition of the local types of modules. Suppose $\{ X_ i \to X\} $ is a covering. Then for every object $U$ of $\mathcal{C}$ we get an induced covering $\{ X_ i \times _ X U \to U\} $. Moreover, the global property for $\mathcal{F}|_{\mathcal{C}/X_ i}$ in part (c) implies the corresponding global property for $\mathcal{F}|_{\mathcal{C}/X_ i \times _ X U}$ by Lemma 18.17.2, hence the sheaf has property (a) by definition. We omit the proof of (b) $\Rightarrow $ (a) in case (7). $\square$

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

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. $\square$