The Stacks project

Proposition 25.10.2. 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 25.10.1. 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 $\coprod _{i \in I} h_{U_ i} \to h_ U$ 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 $\coprod h_{U_ i} \to *$ 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 25.3.4. 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 25.10.1 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 25.6.1. 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 25.6.1 and the fact that weak coverings in $\mathcal{C}$ give coverings in $\mathcal{C}'$. By Theorem 25.10.1 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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