Lemma 18.23.2. Any of the properties (1) – (8) of Definition 18.23.1 is intrinsic (see discussion in Section 18.18).

**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. $\square$

## 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)