The Stacks project

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. $\square$

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