\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*}

The Stacks project

7.38 Criterion for existence of points

This section corresponds to Deligne's appendix to [Exposé VI, SGA4]. In fact it is almost literally the same.

Let $\mathcal{C}$ be a site. Suppose that $(I, \geq )$ is a directed set, and that $(U_ i, f_{ii'})$ is an inverse system over $I$, see Categories, Definition 4.21.2. Given the data $(I, \geq , U_ i, f_{ii'})$ we define

\[ u : \mathcal{C} \longrightarrow \textit{Sets}, \quad u(V) = \mathop{\mathrm{colim}}\nolimits _ i \mathop{Mor}\nolimits _\mathcal {C}(U_ i , V) \]

Let $\mathcal{F} \mapsto \mathcal{F}_ p$ be the stalk functor associated to $u$ as in Section 7.31. It is direct from the definition that actually

\[ \mathcal{F}_ p = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}(U_ i) \]

in this special case. Note that $u$ commutes with all finite limits (I mean those that are representable in $\mathcal{C}$) because each of the functors $V \mapsto \mathop{Mor}\nolimits _\mathcal {C}(U_ i , V)$ do, see Categories, Lemma 4.19.2.

We say that a system $(I, \geq , U_ i, f_{ii'})$ is a refinement of $(J, \geq , V_ j, g_{jj'})$ if $J \subset I$, the ordering on $J$ induced from that of $I$ and $V_ j = U_ j$, $g_{jj'} = f_{jj'}$ (in words, the inverse system over $J$ is induced by that over $I$). Let $u$ be the functor associated to $(I, \geq , U_ i, f_{ii'})$ and let $u'$ be the functor associated to $(J, \geq , V_ j, g_{jj'})$. This induces a transformation of functors

\[ u' \longrightarrow u \]

simply because the colimits for $u'$ are over a subsystem of the systems in the colimits for $u$. In particular we get an associated transformation of stalk functors $\mathcal{F}_{p'} \to \mathcal{F}_ p$, see Lemma 7.36.1.

Lemma 7.38.1. Let $\mathcal{C}$ be a site. Let $(J, \geq , V_ j, g_{jj'})$ be a system as above with associated pair of functors $(u', p')$. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Let $s, s' \in \mathcal{F}_{p'}$ be distinct elements. Let $\{ W_ k \to W\} $ be a finite covering of $\mathcal{C}$. Let $f \in u'(W)$. There exists a refinement $(I, \geq , U_ i, f_{ii'})$ of $(J, \geq , V_ j, g_{jj'})$ such that $s, s'$ map to distinct elements of $\mathcal{F}_ p$ and that the image of $f$ in $u(W)$ is in the image of one of the $u(W_ k)$.

Proof. There exists a $j_0 \in J$ such that $f$ is defined by $f' : V_{j_0} \to W$. For $j \geq j_0$ we set $V_{j, k} = V_ j \times _{f'\circ f_{j j_0}, W} W_ k$. Then $\{ V_{j, k} \to V_ j\} $ is a finite covering in the site $\mathcal{C}$. Hence $\mathcal{F}(V_ j) \subset \prod _ k \mathcal{F}(V_{j, k})$. By Categories, Lemma 4.19.2 once again we see that

\[ \mathcal{F}_{p'} = \mathop{\mathrm{colim}}\nolimits _ j \mathcal{F}(V_ j) \longrightarrow \prod \nolimits _ k \mathop{\mathrm{colim}}\nolimits _ j \mathcal{F}(V_{j, k}) \]

is injective. Hence there exists a $k$ such that $s$ and $s'$ have distinct image in $\mathop{\mathrm{colim}}\nolimits _ j \mathcal{F}(V_{j, k})$. Let $J_0 = \{ j \in J, j \geq j_0\} $ and $I = J \amalg J_0$. We order $I$ so that no element of the second summand is smaller than any element of the first, but otherwise using the ordering on $J$. If $j \in I$ is in the first summand then we use $V_ j$ and if $j \in I$ is in the second summand then we use $V_{j, k}$. We omit the definition of the transition maps of the inverse system. By the above it follows that $s, s'$ have distinct image in $\mathcal{F}_ p$. Moreover, the restriction of $f'$ to $V_{j, k}$ factors through $W_ k$ by construction. $\square$

Lemma 7.38.2. Let $\mathcal{C}$ be a site. Let $(J, \geq , V_ j, g_{jj'})$ be a system as above with associated pair of functors $(u', p')$. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Let $s, s' \in \mathcal{F}_{p'}$ be distinct elements. There exists a refinement $(I, \geq , U_ i, f_{ii'})$ of $(J, \geq , V_ j, g_{jj'})$ such that $s, s'$ map to distinct elements of $\mathcal{F}_ p$ and such that for every finite covering $\{ W_ k \to W\} $ of the site $\mathcal{C}$, and any $f \in u'(W)$ the image of $f$ in $u(W)$ is in the image of one of the $u(W_ k)$.

Proof. Let $E$ be the set of pairs $(\{ W_ k \to W\} , f\in u'(W))$. Consider pairs $(E' \subset E, (I, \geq , U_ i, f_{ii'}))$ such that

  1. $(I, \geq , U_ i, g_{ii'})$ is a refinement of $(J, \geq , V_ j, g_{jj'})$,

  2. $s, s'$ map to distinct elements of $\mathcal{F}_ p$, and

  3. for every pair $(\{ W_ k \to W\} , f\in u'(W)) \in E'$ we have that the image of $f$ in $u(W)$ is in the image of one of the $u(W_ k)$.

We order such pairs by inclusion in the first factor and by refinement in the second. Denote $\mathcal{S}$ the class of all pairs $(E' \subset E, (I, \geq , U_ i, f_{ii'}))$ as above. We claim that the hypothesis of Zorn's lemma holds for $\mathcal{S}$. Namely, suppose that $(E'_ a, (I_ a, \geq , U_ i, f_{ii'}))_{a \in A}$ is a totally ordered subset of $\mathcal{S}$. Then we can define $E' = \bigcup _{a \in A} E'_ a$ and we can set $I = \bigcup _{a \in A} I_ a$. We claim that the corresponding pair $(E' , (I, \geq , U_ i, f_{ii'}))$ is an element of $\mathcal{S}$. Conditions (1) and (3) are clear. For condition (2) you note that

\[ u = \mathop{\mathrm{colim}}\nolimits _{a \in A} u_ a \text{ and correspondingly } \mathcal{F}_ p = \mathop{\mathrm{colim}}\nolimits _{a \in A} \mathcal{F}_{p_ a} \]

The distinctness of the images of $s, s'$ in this stalk follows from the description of a directed colimit of sets, see Categories, Section 4.19. We will simply write $(E', (I, \ldots )) = \bigcup _{a \in A}(E'_ a, (I_ a, \ldots ))$ in this situation.

OK, so Zorn's Lemma would apply if $\mathcal{S}$ was a set, and this would, combined with Lemma 7.38.1 above easily prove the lemma. It doesn't since $\mathcal{S}$ is a class. In order to circumvent this we choose a well ordering on $E$. For $e \in E$ set $E'_ e = \{ e' \in E \mid e' \leq e\} $. By transfinite induction we construct pairs $(E'_ e, (I_ e, \ldots )) \in \mathcal{S}$ such that $e_1 \leq e_2 \Rightarrow (E'_{e_1}, (I_{e_1}, \ldots )) \leq (E'_{e_2}, (I_{e_2}, \ldots ))$. Let $e \in E$, say $e = (\{ W_ k \to W\} , f\in u'(W))$. If $e$ has a predecessor $e - 1$, then we let $(I_ e, \ldots )$ be a refinement of $(I_{e - 1}, \ldots )$ as in Lemma 7.38.1 with respect to the system $e = (\{ W_ k \to W\} , f\in u'(W))$. If $e$ does not have a predecessor, then we let $(I_ e, \ldots )$ be a refinement of $\bigcup _{e' < e} (I_{e'}, \ldots )$ with respect to the system $e = (\{ W_ k \to W\} , f\in u'(W))$. Finally, the union $\bigcup _{e \in E} I_ e$ will be a solution to the problem posed in the lemma. $\square$

Proposition 7.38.3. Let $\mathcal{C}$ be a site. Assume that

  1. finite limits exist in $\mathcal{C}$, and

  2. every covering $\{ U_ i \to U\} _{i \in I}$ has a refinement by a finite covering of $\mathcal{C}$.

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

Proof. We have to show that given any sheaf $\mathcal{F}$ on $\mathcal{C}$, any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and any distinct sections $s, s' \in \mathcal{F}(U)$, there exists a point $p$ such that $s, s'$ have distinct image in $\mathcal{F}_ p$. See Lemma 7.37.3. Consider the system $(J, \geq , V_ j, g_{jj'})$ with $J = \{ 1\} $, $V_1 = U$, $g_{11} = \text{id}_ U$. Apply Lemma 7.38.2. By the result of that lemma we get a system $(I, \geq , U_ i, f_{ii'})$ refining our system such that $s_ p \not= s'_ p$ and such that moreover for every finite covering $\{ W_ k \to W\} $ of the site $\mathcal{C}$ the map $\coprod _ k u(W_ k) \to u(W)$ is surjective. Since every covering of $\mathcal{C}$ can be refined by a finite covering we conclude that $\coprod _ k u(W_ k) \to u(W)$ is surjective for any covering $\{ W_ k \to W\} $ of the site $\mathcal{C}$. This implies that $u = p$ is a point, see Proposition 7.32.2 (and the discussion at the beginning of this section which guarantees that $u$ commutes with finite limits). $\square$

Lemma 7.38.4. Let $\mathcal{C}$ be a site. Let $I$ be a set and for $i \in I$ let $U_ i$ be an object of $\mathcal{C}$ such that

  1. $\coprod h_{U_ i}$ surjects onto the final object of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, and

  2. $\mathcal{C}/U_ i$ satisfies the hypotheses of Proposition 7.38.3.

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

Proof. By assumption (2) and the proposition $\mathcal{C}/U_ i$ has enough points. The points of $\mathcal{C}/U_ i$ give points of $\mathcal{C}$ via the procedure of Lemma 7.33.1. Thus it suffices to show: if $\phi : \mathcal{F} \to \mathcal{G}$ is a map of sheaves on $\mathcal{C}$ such that $\phi |_{\mathcal{C}/U_ i}$ is an isomorphism for all $i$, then $\phi $ is an isomorphism. By assumption (1) for every object $W$ of $\mathcal{C}$ there is a covering $\{ W_ j \to W\} _{j \in J}$ such that for $j \in J$ there is an $i \in I$ and a morphism $f_ j : W_ j \to U_ i$. Then the maps $\mathcal{F}(W_ j) \to \mathcal{G}(W_ j)$ are bijective and similarly for $\mathcal{F}(W_ j \times _ W W_{j'}) \to \mathcal{G}(W_ j \times _ W W_{j'})$. The sheaf condition tells us that $\mathcal{F}(W) \to \mathcal{G}(W)$ is bijective as desired. $\square$


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