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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09VZ. Beware of the difference between the letter 'O' and the digit '0'.