Lemma 7.39.1. 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).
7.39 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
Let \mathcal{F} \mapsto \mathcal{F}_ p be the stalk functor associated to u as in Section 7.32. It is direct from the definition that actually
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{\mathrm{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
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 7.37.1.
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
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 7.39.2. 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
(I, \geq , U_ i, g_{ii'}) is a refinement of (J, \geq , V_ j, g_{jj'}),
s, s' map to distinct elements of \mathcal{F}_ p, and
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
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 7.39.1 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\} . Using transfinite recursion 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 7.39.1 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 7.39.3.reference Let \mathcal{C} be a site. Assume that
finite limits exist in \mathcal{C}, and
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 7.38.3. Consider the system (J, \geq , V_ j, g_{jj'}) with J = \{ 1\} , V_1 = U, g_{11} = \text{id}_ U. Apply Lemma 7.39.2. 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 7.33.3 (and the discussion at the beginning of this section which guarantees that u commutes with finite limits). \square
Lemma 7.39.4. 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
\coprod h_{U_ i} surjects onto the final object of \mathop{\mathit{Sh}}\nolimits (\mathcal{C}), and
\mathcal{C}/U_ i satisfies the hypotheses of Proposition 7.39.3.
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 7.34.2. 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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