The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

7.38 Sites with enough points

Definition 7.38.1. Let $\mathcal{C}$ be a site.

  1. A family of points $\{ p_ i\} _{i\in I}$ is called conservative if every map of sheaves $\phi : \mathcal{F} \to \mathcal{G}$ which is an isomorphism on all the fibres $\mathcal{F}_{p_ i} \to \mathcal{G}_{p_ i}$ is an isomorphism.

  2. We say that $\mathcal{C}$ has enough points if there exists a conservative family of points.

It turns out that you can then check “exactness” at the stalks.

Lemma 7.38.2. Let $\mathcal{C}$ be a site and let $\{ p_ i\} _{i\in I}$ be a conservative family of points. Then

  1. Given any map of sheaves $\varphi : \mathcal{F} \to \mathcal{G}$ we have $\forall i, \varphi _{p_ i}$ injective implies $\varphi $ injective.

  2. Given any map of sheaves $\varphi : \mathcal{F} \to \mathcal{G}$ we have $\forall i, \varphi _{p_ i}$ surjective implies $\varphi $ surjective.

  3. Given any pair of maps of sheaves $\varphi _1, \varphi _2 : \mathcal{F} \to \mathcal{G}$ we have $\forall i, \varphi _{1, p_ i} = \varphi _{2, p_ i}$ implies $\varphi _1 = \varphi _2$.

  4. Given a finite diagram $\mathcal{G} : \mathcal{J} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, a sheaf $\mathcal{F}$ and morphisms $q_ j : \mathcal{F} \to \mathcal{G}_ j$ then $(\mathcal{F}, q_ j)$ is a limit of the diagram if and only if for each $i$ the stalk $(\mathcal{F}_{p_ i}, (q_ j)_{p_ i})$ is one.

  5. Given a finite diagram $\mathcal{F} : \mathcal{J} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, a sheaf $\mathcal{G}$ and morphisms $e_ j : \mathcal{F}_ j \to \mathcal{G}$ then $(\mathcal{G}, e_ j)$ is a colimit of the diagram if and only if for each $i$ the stalk $(\mathcal{G}_{p_ i}, (e_ j)_{p_ i})$ is one.

Proof. We will use over and over again that all the stalk functors commute with any finite limits and colimits and hence with products, fibred products, etc. We will also use that injective maps are the monomorphisms and the surjective maps are the epimorphisms. A map of sheaves $\varphi : \mathcal{F} \to \mathcal{G}$ is injective if and only if $\mathcal{F} \to \mathcal{F} \times _\mathcal {G}\mathcal{F}$ is an isomorphism. Hence (1). Similarly, $\varphi : \mathcal{F} \to \mathcal{G}$ is surjective if and only if $\mathcal{G} \amalg _\mathcal {F} \mathcal{G} \to \mathcal{G}$ is an isomorphism. Hence (2). The maps $a, b : \mathcal{F} \to \mathcal{G}$ are equal if and only if $\mathcal{F} \times _{a, \mathcal{G}, b}\mathcal{F} \to \mathcal{F} \times \mathcal{F}$ is an isomorphism. Hence (3). The assertions (4) and (5) follow immediately from the definitions and the remarks at the start of this proof. $\square$

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. $\square$

In the following lemma the points $q_{i, x}$ are exactly all the points of $\mathcal{C}/U$ lying over the point $p_ i$ according to Lemma 7.35.2.

Lemma 7.38.4. Let $\mathcal{C}$ be a site. Let $U$ be an object of $\mathcal{C}$. let $\{ (p_ i, u_ i)\} _{i\in I}$ be a family of points of $\mathcal{C}$. For $x \in u_ i(U)$ let $q_{i, x}$ be the point of $\mathcal{C}/U$ constructed in Lemma 7.35.1. If $\{ p_ i\} $ is a conservative family of points, then $\{ q_{i, x}\} _{i \in I, x \in u_ i(U)}$ is a conservative family of points of $\mathcal{C}/U$. In particular, if $\mathcal{C}$ has enough points, then so does every localization $\mathcal{C}/U$.

Proof. We know that $j_{U!}$ induces an equivalence $j_{U!} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $, see Lemma 7.25.4. Moreover, we know that $(j_{U!}\mathcal{G})_{p_ i} = \coprod _ x \mathcal{G}_{q_{i, x}}$, see Lemma 7.35.3. Hence the result follows formally. $\square$

The following lemma tells us we can check the existence of points locally on the site.

Lemma 7.38.5. Let $\mathcal{C}$ be a site. Let $\{ U_ i\} _{i \in I}$ be a family of objects of $\mathcal{C}$. Assume

  1. $\coprod h_{U_ i}^\# \to *$ is a surjective map of sheaves, and

  2. each localization $\mathcal{C}/U_ i$ has enough points.

Then $\mathcal{C}$ has enough points.

Proof. For each $i \in I$ let $\{ p_ j\} _{j \in J_ i}$ be a conservative family of points of $\mathcal{C}/U_ i$. For $j \in J_ i$ denote $q_ j : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ the composition of $p_ j$ with the localization morphism $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. Then $q_ j$ is a point, see Lemma 7.34.2. We claim that the family of points $\{ q_ j\} _{j \in \coprod J_ i}$ is conservative. Namely, let $\mathcal{F} \to \mathcal{G}$ be a map of sheaves on $\mathcal{C}$ such that $\mathcal{F}_{q_ j} \to \mathcal{G}_{q_ j}$ is an isomorphism for all $j \in \coprod J_ i$. Let $W$ be an object of $\mathcal{C}$. By assumption (1) there exists a covering $\{ W_ a \to W\} $ and morphisms $W_ a \to U_{i(a)}$. Since $(\mathcal{F}|_{\mathcal{C}/U_{i(a)}})_{p_ j} = \mathcal{F}_{q_ j}$ and $(\mathcal{G}|_{\mathcal{C}/U_{i(a)}})_{p_ j} = \mathcal{G}_{q_ j}$ by Lemma 7.34.2 we see that $\mathcal{F}|_{U_{i(a)}} \to \mathcal{G}|_{U_{i(a)}}$ is an isomorphism since the family of points $\{ p_ j\} _{j \in J_{i(a)}}$ is conservative. Hence $\mathcal{F}(W_ a) \to \mathcal{G}(W_ a)$ is bijective for each $a$. Similarly $\mathcal{F}(W_ a \times _ W W_ b) \to \mathcal{G}(W_ a \times _ W W_ b)$ is bijective for each $a, b$. By the sheaf condition this shows that $\mathcal{F}(W) \to \mathcal{G}(W)$ is bijective, i.e., $\mathcal{F} \to \mathcal{G}$ is an isomorphism. $\square$


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