An object $U$ of a site is weakly contractible if every surjection $\mathcal{F} \to \mathcal{G}$ of sheaves of sets gives rise to a surjection $\mathcal{F}(U) \to \mathcal{G}(U)$, see Sites, Definition 7.40.2.
Lemma 21.51.1. Let $\mathcal{C}$ be a site. Let $U$ be a weakly contractible object of $\mathcal{C}$. Then
the functor $\mathcal{F} \mapsto \mathcal{F}(U)$ is an exact functor $\textit{Ab}(\mathcal{C}) \to \textit{Ab}$,
$H^ p(U, \mathcal{F}) = 0$ for every abelian sheaf $\mathcal{F}$ and all $p \geq 1$, and
for any sheaf of groups $\mathcal{G}$ any $\mathcal{G}$-torsor has a section over $U$.
Proof. The first statement follows immediately from the definition (see also Homology, Section 12.7). The higher derived functors vanish by Derived Categories, Lemma 13.16.9. Let $\mathcal{F}$ be a $\mathcal{G}$-torsor. Then $\mathcal{F} \to *$ is a surjective map of sheaves. Hence (3) follows from the definition as well. $\square$
It is convenient to list some consequences of having enough weakly contractible objects here.
Proposition 21.51.2. Let $\mathcal{C}$ be a site. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ such that every $U \in \mathcal{B}$ is weakly contractible and every object of $\mathcal{C}$ has a covering by elements of $\mathcal{B}$. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Then
A complex $\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3$ of $\mathcal{O}$-modules is exact, if and only if $\mathcal{F}_1(U) \to \mathcal{F}_2(U) \to \mathcal{F}_3(U)$ is exact for all $U \in \mathcal{B}$.
Every object $K$ of $D(\mathcal{O})$ is a derived limit of its canonical truncations: $K = R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n} K$.
Given an inverse system $\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1$ with surjective transition maps, the projection $\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n \to \mathcal{F}_1$ is surjective.
Products are exact on $\textit{Mod}(\mathcal{O})$.
Products on $D(\mathcal{O})$ can be computed by taking products of any representative complexes.
If $(\mathcal{F}_ n)$ is an inverse system of $\mathcal{O}$-modules, then $R^ p\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n = 0$ for all $p > 1$ and
where the map is $(x_ n) \mapsto (x_ n - f(x_{n + 1}))$.
If $(K_ n)$ is an inverse system of objects of $D(\mathcal{O})$, then there are short exact sequences
\[ R^1\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n = \mathop{\mathrm{Coker}}(\prod \mathcal{F}_ n \to \prod \mathcal{F}_ n) \]
\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(K_ n) \to H^ p(R\mathop{\mathrm{lim}}\nolimits K_ n) \to \mathop{\mathrm{lim}}\nolimits H^ p(K_ n) \to 0 \]
Proof. Proof of (1). If the sequence is exact, then evaluating at any weakly contractible element of $\mathcal{C}$ gives an exact sequence by Lemma 21.51.1. Conversely, assume that $\mathcal{F}_1(U) \to \mathcal{F}_2(U) \to \mathcal{F}_3(U)$ is exact for all $U \in \mathcal{B}$. Let $V$ be an object of $\mathcal{C}$ and let $s \in \mathcal{F}_2(V)$ be an element of the kernel of $\mathcal{F}_2 \to \mathcal{F}_3$. By assumption there exists a covering $\{ U_ i \to V\} $ with $U_ i \in \mathcal{B}$. Then $s|_{U_ i}$ lifts to a section $s_ i \in \mathcal{F}_1(U_ i)$. Thus $s$ is a section of the image sheaf $\mathop{\mathrm{Im}}(\mathcal{F}_1 \to \mathcal{F}_2)$. In other words, the sequence $\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3$ is exact.
Proof of (2). This holds by Lemma 21.23.10 with $d = 0$.
Proof of (3). Let $(\mathcal{F}_ n)$ be a system as in (2) and set $\mathcal{F} = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n$. If $U \in \mathcal{B}$, then $\mathcal{F}(U) = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n(U)$ surjects onto $\mathcal{F}_1(U)$ as all the transition maps $\mathcal{F}_{n + 1}(U) \to \mathcal{F}_ n(U)$ are surjective. Thus $\mathcal{F} \to \mathcal{F}_1$ is surjective by Sites, Definition 7.11.1 and the assumption that every object has a covering by elements of $\mathcal{B}$.
Proof of (4). Let $\mathcal{F}_{i, 1} \to \mathcal{F}_{i, 2} \to \mathcal{F}_{i, 3}$ be a family of exact sequences of $\mathcal{O}$-modules. We want to show that $\prod \mathcal{F}_{i, 1} \to \prod \mathcal{F}_{i, 2} \to \prod \mathcal{F}_{i, 3}$ is exact. We use the criterion of (1). Let $U \in \mathcal{B}$. Then
is the same as
Each of the sequences $\mathcal{F}_{i, 1}(U) \to \mathcal{F}_{i, 2}(U) \to \mathcal{F}_{i, 3}(U)$ are exact by (1). Thus the displayed sequences are exact by Homology, Lemma 12.32.1. We conclude by (1) again.
Proof of (5). Follows from (4) and (slightly generalized) Derived Categories, Lemma 13.34.2.
Proof of (6) and (7). We refer to Section 21.23 for a discussion of derived and homotopy limits and their relationship. By Derived Categories, Definition 13.34.1 we have a distinguished triangle
Taking the long exact sequence of cohomology sheaves we obtain
Since products are exact by (4) this becomes
Now we first apply this to the case $K_ n = \mathcal{F}_ n[0]$ where $(\mathcal{F}_ n)$ is as in (6). We conclude that (6) holds. Next we apply it to $(K_ n)$ as in (7) and we conclude (7) holds. $\square$
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