Lemma 7.38.3. Let $\mathcal{C}$ be a site and let $\{ (p_ i, u_ i)\} _{i\in I}$ be a family of points. The family is conservative if and only if for every sheaf $\mathcal{F}$ and every $U\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and every pair of distinct sections $s, s' \in \mathcal{F}(U)$, $s \not= s'$ there exists an $i$ and $x\in u_ i(U)$ such that the triples $(U, x, s)$ and $(U, x, s')$ define distinct elements of $\mathcal{F}_{p_ i}$.

**Proof.**
Suppose that the family is conservative and that $\mathcal{F}$, $U$, and $s, s'$ are as in the lemma. The sections $s$, $s'$ define maps $a, a' : (h_ U)^\# \to \mathcal{F}$ which are distinct. Hence, by Lemma 7.38.2 there is an $i$ such that $a_{p_ i} \not= a'_{p_ i}$. Recall that $(h_ U)^\# _{p_ i} = u_ i(U)$, by Lemmas 7.32.3 and 7.32.5. Hence there exists an $x \in u_ i(U)$ such that $a_{p_ i}(x) \not= a'_{p_ i}(x)$ in $\mathcal{F}_{p_ i}$. Unwinding the definitions you see that $(U, x, s)$ and $(U, x, s')$ are as in the statement of the lemma.

To prove the converse, assume the condition on the existence of points of the lemma. Let $\phi : \mathcal{F} \to \mathcal{G}$ be a map of sheaves which is an isomorphism at all the stalks. We have to show that $\phi $ is both injective and surjective, see Lemma 7.11.2. Injectivity is an immediate consequence of the assumption. To show surjectivity we have to show that $\mathcal{G} \amalg _\mathcal {F} \mathcal{G} \to \mathcal{G}$ is an isomorphism (Categories, Lemma 4.13.3). Since this map is clearly surjective, it suffices to check injectivity which follows as $\mathcal{G} \amalg _\mathcal {F} \mathcal{G} \to \mathcal{G}$ is injective on all stalks by assumption. $\square$

## 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Your email address will not be published.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)