Theorem 25.10.1. Let \mathcal{C} be a site with fibre products. Let X be an object of \mathcal{C}. Let i \geq 0. The functors
are canonically isomorphic.
Theorem 25.10.1. Let \mathcal{C} be a site with fibre products. Let X be an object of \mathcal{C}. Let i \geq 0. The functors
are canonically isomorphic.
Proof using spectral sequences.. Suppose that \xi \in H^ p(X, \mathcal{F}) for some p \geq 0. Let us show that \xi is in the image of the map \check{H}^ p(X, \mathcal{F}) \to H^ p(X, \mathcal{F}) of Lemma 25.5.3 for some hypercovering K of X.
This is true if p = 0 by Lemma 25.5.1. If p = 1, choose a Čech hypercovering K of X as in Example 25.3.4 starting with a covering K_0 = \{ U_ i \to X\} in the site \mathcal{C} such that \xi |_{U_ i} = 0, see Cohomology on Sites, Lemma 21.7.3. It follows immediately from the spectral sequence in Lemma 25.5.3 that \xi comes from an element of \check{H}^1(K, \mathcal{F}) in this case. In general, choose any hypercovering K of X such that \xi maps to zero in \underline{H}^ p(\mathcal{F})(K_0) (using Example 25.3.4 and Cohomology on Sites, Lemma 21.7.3 again). By the spectral sequence of Lemma 25.5.3 the obstruction for \xi to come from an element of \check{H}^ p(K, \mathcal{F}) is a sequence of elements \xi _1, \ldots , \xi _{p - 1} with \xi _ q \in \check{H}^{p - q}(K, \underline{H}^ q(\mathcal{F})) (more precisely the images of the \xi _ q in certain subquotients of these groups).
We can inductively replace the hypercovering K by refinements such that the obstructions \xi _1, \ldots , \xi _{p - 1} restrict to zero (and not just the images in the subquotients – so no subtlety here). Indeed, suppose we have already managed to reach the situation where \xi _{q + 1}, \ldots , \xi _{p - 1} are zero. Note that \xi _ q \in \check{H}^{p - q}(K, \underline{H}^ q(\mathcal{F})) is the class of some element
if K_{p - q} = \{ U_ i \to X\} _{i \in I}. Let \xi _{q, i} be the component of \tilde\xi _ q in H^ q(U_ i, \mathcal{F}). As q \geq 1 we can use Cohomology on Sites, Lemma 21.7.3 yet again to choose coverings \{ U_{i, j} \to U_ i\} of the site such that each restriction \xi _{q, i}|_{U_{i, j}} = 0. Consider the object Z = \{ U_{i, j} \to X\} of the category \text{SR}(\mathcal{C}, X) and its obvious morphism u : Z \to K_{p - q}. It is clear that u is a covering, see Definition 25.3.1. By Lemma 25.7.3 there exists a morphism L \to K of hypercoverings of X such that L_{p - q} \to K_{p - q} factors through u. Then clearly the image of \xi _ q in \underline{H}^ q(\mathcal{F})(L_{p - q}). is zero. Since the spectral sequence of Lemma 25.5.3 is functorial this means that after replacing K by L we reach the situation where \xi _ q, \ldots , \xi _{p - 1} are all zero. Continuing like this we end up with a hypercovering where they are all zero and hence \xi is in the image of the map \check{H}^ p(X, \mathcal{F}) \to H^ p(X, \mathcal{F}).
Suppose that K is a hypercovering of X, that \xi \in \check{H}^ p(K, \mathcal{F}) and that the image of \xi under the map \check{H}^ p(X, \mathcal{F}) \to H^ p(X, \mathcal{F}) of Lemma 25.5.3 is zero. To finish the proof of the theorem we have to show that there exists a morphism of hypercoverings L \to K such that \xi restricts to zero in \check{H}^ p(L, \mathcal{F}). By the spectral sequence of Lemma 25.5.3 the vanishing of the image of \xi in H^ p(X, \mathcal{F}) means that there exist elements \xi _1, \ldots , \xi _{p - 2} with \xi _ q \in \check{H}^{p - 1 - q}(K, \underline{H}^ q(\mathcal{F})) (more precisely the images of these in certain subquotients) such that the images d_{q + 1}^{p - 1 - q, q}\xi _ q (in the spectral sequence) add up to \xi . Hence by exactly the same mechanism as above we can find a morphism of hypercoverings L \to K such that the restrictions of the elements \xi _ q, q = 1, \ldots , p - 2 in \check{H}^{p - 1 - q}(L, \underline{H}^ q(\mathcal{F})) are zero. Then it follows that \xi is zero since the morphism L \to K induces a morphism of spectral sequences according to Lemma 25.5.3. \square
Proof without using spectral sequences.. We have seen the result for i = 0, see Lemma 25.5.1. We know that the functors H^ i(X, -) form a universal \delta -functor, see Derived Categories, Lemma 13.20.4. In order to prove the theorem it suffices to show that the sequence of functors \check{H}^ i_{HC}(X, -) forms a \delta -functor. Namely we know that Čech cohomology is zero on injective sheaves (Lemma 25.5.2) and then we can apply Homology, Lemma 12.12.4.
Let
be a short exact sequence of abelian sheaves on \mathcal{C}. Let \xi \in \check{H}^ p_{HC}(X, \mathcal{H}). Choose a hypercovering K of X and an element \sigma \in \mathcal{H}(K_ p) representing \xi in cohomology. There is a corresponding exact sequence of complexes
but we are not assured that there is a zero on the right also and this is the only thing that prevents us from defining \delta (\xi ) by a simple application of the snake lemma. Recall that
if K_ p = \{ U_ i \to X\} . Let \sigma =\prod \sigma _ i with \sigma _ i \in \mathcal{H}(U_ i). Since \mathcal{G} \to \mathcal{H} is a surjection of sheaves we see that there exist coverings \{ U_{i, j} \to U_ i\} such that \sigma _ i|_{U_{i, j}} is the image of some element \tau _{i, j} \in \mathcal{G}(U_{i, j}). Consider the object Z = \{ U_{i, j} \to X\} of the category \text{SR}(\mathcal{C}, X) and its obvious morphism u : Z \to K_ p. It is clear that u is a covering, see Definition 25.3.1. By Lemma 25.7.3 there exists a morphism L \to K of hypercoverings of X such that L_ p \to K_ p factors through u. After replacing K by L we may therefore assume that \sigma is the image of an element \tau \in \mathcal{G}(K_ p). Note that d(\sigma ) = 0, but not necessarily d(\tau ) = 0. Thus d(\tau ) \in \mathcal{F}(K_{p + 1}) is a cocycle. In this situation we define \delta (\xi ) as the class of the cocycle d(\tau ) in \check{H}^{p + 1}_{HC}(X, \mathcal{F}).
At this point there are several things to verify: (a) \delta (\xi ) does not depend on the choice of \tau , (b) \delta (\xi ) does not depend on the choice of the hypercovering L \to K such that \sigma lifts, and (c) \delta (\xi ) does not depend on the initial hypercovering and \sigma chosen to represent \xi . We omit the verification of (a), (b), and (c); the independence of the choices of the hypercoverings really comes down to Lemmas 25.7.2 and 25.9.2. We also omit the verification that \delta is functorial with respect to morphisms of short exact sequences of abelian sheaves on \mathcal{C}.
Finally, we have to verify that with this definition of \delta our short exact sequence of abelian sheaves above leads to a long exact sequence of Čech cohomology groups. First we show that if \delta (\xi ) = 0 (with \xi as above) then \xi is the image of some element \xi ' \in \check{H}^ p_{HC}(X, \mathcal{G}). Namely, if \delta (\xi ) = 0, then, with notation as above, we see that the class of d(\tau ) is zero in \check{H}^{p + 1}_{HC}(X, \mathcal{F}). Hence there exists a morphism of hypercoverings L \to K such that the restriction of d(\tau ) to an element of \mathcal{F}(L_{p + 1}) is equal to d(\upsilon ) for some \upsilon \in \mathcal{F}(L_ p). This implies that \tau |_{L_ p} + \upsilon form a cocycle, and determine a class \xi ' \in \check{H}^ p(L, \mathcal{G}) which maps to \xi as desired.
We omit the proof that if \xi ' \in \check{H}^{p + 1}_{HC}(X, \mathcal{F}) maps to zero in \check{H}^{p + 1}_{HC}(X, \mathcal{G}), then it is equal to \delta (\xi ) for some \xi \in \check{H}^ p_{HC}(X, \mathcal{H}). \square
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