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7.38. Criterion for existence of points

This section corresponds to Deligne's appendix to [SGA4, Exposé VI]. 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 $$ u : \mathcal{C} \longrightarrow \textit{Sets}, \quad u(V) = \mathop{\rm colim}\nolimits_i \mathop{\rm Mor}\nolimits_\mathcal{C}(U_i , V) $$ 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 $$ \mathcal{F}_p = \mathop{\rm colim}\nolimits_i \mathcal{F}(U_i) $$ 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 $V \mapsto \mathop{\rm Mor}\nolimits_\mathcal{C}(U_i , V)$ 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 $$ \mathcal{F}_{p'} = \mathop{\rm colim}\nolimits_j \mathcal{F}(V_j) \longrightarrow \prod\nolimits_k \mathop{\rm colim}\nolimits_j \mathcal{F}(V_{j, k}) $$ is injective. Hence there exists a $k$ such that $s$ and $s'$ have distinct image in $\mathop{\rm colim}\nolimits_j \mathcal{F}(V_{j, k})$. 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 $$ u = \mathop{\rm colim}\nolimits_{a \in A} u_a \text{ and correspondingly } \mathcal{F}_p = \mathop{\rm colim}\nolimits_{a \in A} \mathcal{F}_{p_a} $$ 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 $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$, 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 $\mathop{\textit{Sh}}\nolimits(\mathcal{C})$, 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. $\square$

    The code snippet corresponding to this tag is a part of the file sites.tex and is located in lines 8305–8535 (see updates for more information).

    \section{Criterion for existence of points}
    \label{section-criterion-points}
    
    \noindent
    This section corresponds to Deligne's appendix to \cite[Expos\'e VI]{SGA4}.
    In fact it is almost literally the same.
    
    \medskip\noindent
    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 \ref{categories-definition-system-over-poset}.
    Given the data $(I, \geq, U_i, f_{ii'})$ we define
    $$
    u : \mathcal{C} \longrightarrow \textit{Sets}, \quad
    u(V) = \colim_i \Mor_\mathcal{C}(U_i , V)
    $$
    Let $\mathcal{F} \mapsto \mathcal{F}_p$ be the stalk functor
    associated to $u$ as in Section \ref{section-points}.
    It is direct from the definition that actually
    $$
    \mathcal{F}_p = \colim_i \mathcal{F}(U_i)
    $$
    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 $V \mapsto \Mor_\mathcal{C}(U_i , V)$
    do, see Categories, Lemma \ref{categories-lemma-directed-commutes}.
    
    \medskip\noindent
    We say that a system $(I, \geq, U_i, f_{ii'})$
    is a {\it 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 \ref{lemma-maps-u-points}.
    
    \begin{lemma}
    \label{lemma-refine}
    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 $u(W_k)$.
    \end{lemma}
    
    \begin{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 \ref{categories-lemma-directed-commutes}
    once again we see that
    $$
    \mathcal{F}_{p'} =
    \colim_j \mathcal{F}(V_j)
    \longrightarrow
    \prod\nolimits_k \colim_j \mathcal{F}(V_{j, k})
    $$
    is injective. Hence there exists a $k$ such that $s$ and $s'$
    have distinct image in $\colim_j \mathcal{F}(V_{j, k})$.
    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.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-refine-all-at-once}
    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 $u(W_k)$.
    \end{lemma}
    
    \begin{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
    \begin{enumerate}
    \item $(I, \geq, U_i, g_{ii'})$ is a refinement of $(J, \geq, V_j, g_{jj'})$,
    \item $s, s'$ map to distinct elements of $\mathcal{F}_p$, and
    \item 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 $u(W_k)$.
    \end{enumerate}
    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
    $$
    u = \colim_{a \in A} u_a
    \text{ and correspondingly }
    \mathcal{F}_p = \colim_{a \in A} \mathcal{F}_{p_a}
    $$
    The distinctness of the images of $s, s'$ in this stalk follows
    from the description of a directed colimit of sets, see
    Categories, Section \ref{categories-section-directed-colimits}.
    We will simply write
    $(E', (I, \ldots)) = \bigcup_{a \in A}(E'_a, (I_a, \ldots))$
    in this situation.
    
    \medskip\noindent
    OK, so Zorn's Lemma would apply if $\mathcal{S}$ was a set,
    and this would, combined with Lemma \ref{lemma-refine} 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 \ref{lemma-refine} 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.
    \end{proof}
    
    \begin{proposition}
    \label{proposition-criterion-points}
    Let $\mathcal{C}$ be a site. Assume that
    \begin{enumerate}
    \item finite limits exist in $\mathcal{C}$, and
    \item every covering $\{U_i \to U\}_{i \in I}$
    has a refinement by a finite covering of $\mathcal{C}$.
    \end{enumerate}
    Then $\mathcal{C}$ has enough points.
    \end{proposition}
    
    \begin{proof}
    We have to show that given any sheaf
    $\mathcal{F}$ on $\mathcal{C}$, any $U \in \Ob(\mathcal{C})$,
    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 \ref{lemma-enough}.
    Consider the system $(J, \geq, V_j, g_{jj'})$
    with $J = \{1\}$, $V_1 = U$, $g_{11} = \text{id}_U$.
    Apply Lemma \ref{lemma-refine-all-at-once}.
    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 {\it any}
    covering $\{W_k \to W\}$ of the site $\mathcal{C}$.
    This implies that $u = p$ is a point, see
    Proposition \ref{proposition-point-limits} (and the discussion
    at the beginning of this section which guarantees that $u$
    commutes with finite limits).
    \end{proof}
    
    \begin{lemma}
    \label{lemma-criterion-points}
    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
    \begin{enumerate}
    \item $\coprod h_{U_i}$ surjects onto
    the final object of $\Sh(\mathcal{C})$, and
    \item $\mathcal{C}/U_i$ satisfies the hypotheses of
    Proposition \ref{proposition-criterion-points}.
    \end{enumerate}
    Then $\mathcal{C}$ has enough points.
    \end{lemma}
    
    \begin{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 \ref{lemma-point-morphism-sites}.
    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.
    \end{proof}

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