The Stacks project

Theorem 24.10.1. Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $i \geq 0$. The functors

\begin{eqnarray*} \textit{Ab}(\mathcal{C}) & \longrightarrow & \textit{Ab} \\ \mathcal{F} & \longmapsto & H^ i(X, \mathcal{F}) \\ \mathcal{F} & \longmapsto & \check{H}^ i_{\text{HC}}(X, \mathcal{F}) \end{eqnarray*}

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 24.5.3 for some hypercovering $K$ of $X$.

This is true if $p = 0$ by Lemma 24.5.1. If $p = 1$, choose a Čech hypercovering $K$ of $X$ as in Example 24.3.4 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 21.8.3. It follows immediately from the spectral sequence in Lemma 24.5.3 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 24.3.4 and Cohomology on Sites, Lemma 21.8.3 again). By the spectral sequence of Lemma 24.5.3 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

\[ \tilde\xi _ q \in \underline{H}^ q(\mathcal{F})(K_{p - q}) = \prod H^ q(U_ i, \mathcal{F}) \]

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 21.8.3 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 24.3.1. By Lemma 24.7.3 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 24.5.3 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 24.5.3 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 24.5.3 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 24.5.3. $\square$

Proof without using spectral sequences.. We have seen the result for $i = 0$, see Lemma 24.5.1. 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 24.5.2) and then we can apply Homology, Lemma 12.11.4.

Let

\[ 0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0 \]

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

\[ 0 \to s(\mathcal{F}(K)) \to s(\mathcal{G}(K)) \to s(\mathcal{H}(K)) \]

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

\[ \mathcal{H}(K_ p) = \prod \mathcal{H}(U_ i) \]

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 24.3.1. By Lemma 24.7.3 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 24.7.2 and 24.9.2. 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 $\xi \in \check{H}^ p_{HC}(X, \mathcal{H})$. $\square$


Comments (2)

Comment #3532 by Kestutis Cesnavicius on

It would be nice to upgrade the statement a little bit by saying that the direct limit of the hypercover Čech complexes is canonically (and, in particular, functorially in ) identified in the derived category with (the limit is along the maps explained in Tag 01GY). The present statement then follows by taking cohomology.

Suggested slogan: Hypercover Čech cohomology computes the derived functor cohomology

Also, I have a general suggestion regarding slogans: to make browsing through the Stacks Project easier and slogans more useful, it would be nice to by default display them in the "Expand all" view of the Chapter (or a section) as kind of titles for what various Tags actually stand for. To illustrate what I mean, if one looks at the table of contents of some relatively short chapter, say, https://stacks.math.columbia.edu/tag/073W, and clicks on "Expand all" one sees a list of the form "Lemma XYZW"; it would be nice to instead have "Lemma XYZW: the slogan associated to XYZW". I see that one can currently hover the mouse over to see the slogan, but I think displaying them by default as kind of titles of Lemmas/Theorems, etc. would be more useful.

Comment #3533 by on

Well, actually the thing you are asking for is tricky, even if it is true. The reason that I say that it is true is explained in Remark 24.9.3. But maybe I should first explain why it is tricky: what is clear is that the system of Cech cohomology groups over the category of hypercoverings is directed (as defined in Definition 4.19.1). This follows from Lemma 24.9.2 and the fact that homotopic maps between simplicial abelian groups define the same map on cohomology. However, what is not clear and what is "usually" not the case, is that the category of hypercoverings is directed. Hence we have no idea what the colimit of the Cech complexes (taken in the abelian category of complexes) has as its cohomology!!! If we work out the suggestion in Remark 24.9.3 then we can take a colimit in a suitable enhanced category (not just the abelian category of complexes, but eg in the corresponding dg category), but this is (currently) outside of the remit of the Stacks project. I assume you can find it worked out in other texts. If somebody has a precise reference (for the exact thing we are discussing here), then I would like to add it to the remark and/or the introduction to this section.

PS: strange things may happen if you look at unbounded complexes, so the above discussion is meant only for bounded below complexes of sheaves of abelian groups, for example the cohomology of a single ablelian sheaf as in the statement.


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