The Stacks project

\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*}

24.10 Cohomology and hypercoverings

Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $\mathcal{F}$ be a sheaf of abelian groups on $\mathcal{C}$. Let $K, L$ be hypercoverings of $X$. If $a, b : K \to L$ are homotopic maps, then $\mathcal{F}(a), \mathcal{F}(b) : \mathcal{F}(K) \to \mathcal{F}(L)$ are homotopic maps, see Simplicial, Lemma 14.28.3. Hence have the same effect on cohomology groups of the associated cochain complexes, see Simplicial, Lemma 14.28.5. We are going to use this to define the colimit over all hypercoverings.

Let us temporarily denote $\text{HC}(\mathcal{C}, X)$ the category of hypercoverings of $X$. We have seen that this is a category and not a “big” category, see Lemma 24.3.5. This will be the index category for our diagram, see Categories, Section 4.14 for notation. Consider the diagram

\[ \check{H}^ i(-, \mathcal{F}) : \text{HC}(\mathcal{C}, X) \longrightarrow \textit{Ab}. \]

By Lemma 24.7.2 and Lemma 24.9.2, and the remark on homotopies above, this diagram is directed, see Categories, Definition 4.19.1. Thus the colimit

\[ \check{H}^ i_{\text{HC}}(X, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits _{K \in \text{HC}(\mathcal{C}, X)} \check{H}^ i(K, \mathcal{F}) \]

has a particularly simple description (see location cited).

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 $\xi \in \check{H}^ p_{HC}(X, \mathcal{H})$. $\square$

Next, we deduce Verdier's case of Theorem 24.10.1 by a sleight of hand.

Proposition 24.10.2. Let $\mathcal{C}$ be a site with fibre products and products of pairs. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$. Let $i \geq 0$. Then

  1. for every $\xi \in H^ i(\mathcal{F})$ there exists a hypercovering $K$ such that $\xi $ is in the image of the canonical map $\check{H}^ i(K, \mathcal{F}) \to H^ i(\mathcal{F})$, and

  2. if $K, L$ are hypercoverings and $\xi _ K \in \check{H}^ i(K, \mathcal{F})$, $\xi _ L \in \check{H}^ i(L, \mathcal{F})$ are elements mapping to the same element of $H^ i(\mathcal{F})$, then there exists a hypercovering $M$ and morphisms $M \to K$ and $M \to L$ such that $\xi _ K$ and $\xi _ L$ map to the same element of $\check{H}^ i(M, \mathcal{F})$.

In other words, modulo set theoretical issues, the cohomology groups of $\mathcal{F}$ on $\mathcal{C}$ are the colimit of the Čech cohomology groups of $\mathcal{F}$ over all hypercoverings.

Proof. This result is a trivial consequence of Theorem 24.10.1. Namely, we can artificially replace $\mathcal{C}$ with a slightly bigger site $\mathcal{C}'$ such that (I) $\mathcal{C}'$ has a final object $X$ and (II) hypercoverings in $\mathcal{C}$ are more or less the same thing as hypercoverings of $X$ in $\mathcal{C}'$. But due to the nature of things, there is quite a bit of bookkeeping to do.

Let us call a family of morphisms $\{ U_ i \to U\} $ in $\mathcal{C}$ with fixed target a weak covering if the sheafification of the map $\prod _{i \in I} h_{U_ i} \to h_ U$ becomes surjective. We construct a new site $\mathcal{C}'$ as follows

  1. as a category set $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C}') = \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) \amalg \{ X\} $ and add a unique morphism to $X$ from every object of $\mathcal{C}'$,

  2. $\mathcal{C}'$ has fibre products as fibre products and products of pairs exist in $\mathcal{C}$,

  3. coverings of $\mathcal{C}'$ are weak coverings of $\mathcal{C}$ together with those $\{ U_ i \to X\} _{i \in I}$ such that either $U_ i = X$ for some $i$, or $U_ i \not= X$ for all $i$ and the map $\prod h_{U_ i} \to *$ of presheaves on $\mathcal{C}$ becomes surjective after sheafification on $\mathcal{C}$,

  4. we apply Sets, Lemma 3.11.1 to restrict the coverings to obtain our site $\mathcal{C}'$.

Then $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}') = \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ because the inclusion functor $\mathcal{C} \to \mathcal{C}'$ is a special cocontinuous functor (see Sites, Definition 7.29.2). We omit the straightforward verifications.

Choose a covering $\{ U_ i \to X\} $ of $\mathcal{C}'$ such that $U_ i$ is an object of $\mathcal{C}$ for all $i$ (possible because $\mathcal{C} \to \mathcal{C}'$ is special cocontinuous). Then $K_0 = \{ U_ i \to X\} $ is a covering in the site $\mathcal{C}'$ constructed above. We view $K_0$ as an object of $\text{SR}(\mathcal{C}', X)$ and we set $K_{init} = \text{cosk}_0(K_0)$. Then $K_{init}$ is a hypercovering of $X$, see Example 24.3.4. Note that every $K_{init, n}$ has the shape $\{ W_ j \to X\} $ with $W_ j \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

Proof of (1). Choose $\xi \in H^ i(\mathcal{F}) = H^ i(X, \mathcal{F}')$ where $\mathcal{F}'$ is the abelian sheaf on $\mathcal{C}'$ corresponding to $\mathcal{F}$ on $\mathcal{C}$. By Theorem 24.10.1 there exists a morphism of hypercoverings $K' \to K_{init}$ of $X$ in $\mathcal{C}'$ such that $\xi $ comes from an element of $\check{H}^ i(K', \mathcal{F})$. Write $K'_ n = \{ U_{n, j} \to X\} $. Now since $K'_ n$ maps to $K_{init, n}$ we see that $U_{n, j}$ is an object of $\mathcal{C}$. Hence we can define a simplicial object $K$ of $\text{SR}(\mathcal{C})$ by setting $K_ n = \{ U_{n, j}\} $. Since coverings in $\mathcal{C}'$ consisting of families of morphisms of $\mathcal{C}$ are weak coverings, we see that $K$ is a hypercovering in the sense of Definition 24.6.1. Finally, since $\mathcal{F}'$ is the unique sheaf on $\mathcal{C}'$ whose restriction to $\mathcal{C}$ is equal to $\mathcal{F}$ we see that the Čech complexes $s(\mathcal{F}(K))$ and $s(\mathcal{F}'(K'))$ are identical and (1) follows. (Compatibility with map into cohomology groups omitted.)

Proof of (2). Let $K$ and $L$ be hypercoverings in $\mathcal{C}$. Let $K'$ and $L'$ be the simplicial objects of $\text{SR}(\mathcal{C}', X)$ gotten from $K$ and $L$ by the functor $\text{SR}(\mathcal{C}) \to \text{SR}(\mathcal{C}', X)$, $\{ U_ i\} \mapsto \{ U_ i \to X\} $. As before we have equality of Čech complexes and hence we obtain $\xi _{K'}$ and $\xi _{L'}$ mapping to the same cohomology class of $\mathcal{F}'$ over $\mathcal{C}'$. After possibly enlarging our choice of coverings in $\mathcal{C}'$ (due to a set theoretical issue) we may assume that $K'$ and $L'$ are hypercoverings of $X$ in $\mathcal{C}'$; this is true by our definition of hypercoverings in Definition 24.6.1 and the fact that weak coverings in $\mathcal{C}$ give coverings in $\mathcal{C}'$. By Theorem 24.10.1 there exists a hypercovering $M'$ of $X$ in $\mathcal{C}'$ and morphisms $M' \to K'$, $M' \to L'$, and $M' \to K_{init}$ such that $\xi _{K'}$ and $\xi _{L'}$ restrict to the same element of $\check{H}^ i(M', \mathcal{F})$. Unwinding this statement as above we find that (2) is true. $\square$


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