Lemma 21.13.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K$ be a presheaf of sets on $\mathcal{C}$. Let $\mathcal{F}$ be an $\mathcal{O}$-module and denote $\mathcal{F}_{ab}$ the underlying sheaf of abelian groups. Then $H^ p(K, \mathcal{F}) = H^ p(K, \mathcal{F}_{ab})$.
21.13 Totally acyclic sheaves
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K$ be a presheaf of sets on $\mathcal{C}$ (we intentionally use a roman capital here to distinguish from abelian sheaves). Given a sheaf of $\mathcal{O}$-modules $\mathcal{F}$ we set
\[ \mathcal{F}(K) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(K, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(K^\# , \mathcal{F}) \]
The functor $\mathcal{F} \mapsto \mathcal{F}(K)$ is a left exact functor $\textit{Mod}(\mathcal{O}) \to \textit{Ab}$ hence we have its right derived functors. We will denote these $H^ p(K, \mathcal{F})$ so that $H^0(K, \mathcal{F}) = \mathcal{F}(K)$.
Here are some observations:
Since $\mathcal{F}(K) = \mathcal{F}(K^\# )$, we have $H^ p(K, \mathcal{F}) = H^ p(K^\# , \mathcal{F})$. Allowing $K$ to be a presheaf in the definition above is a purely notational convenience.
Suppose that $K = h_ U$ or $K = h_ U^\# $ for some object $U$ of $\mathcal{C}$. Then $H^ p(K, \mathcal{F}) = H^ p(U, \mathcal{F})$, because $\mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_ U^\# , \mathcal{F}) = \mathcal{F}(U)$, see Sites, Section 7.12.
If $\mathcal{O} = \mathbf{Z}$ (the constant sheaf), then the cohomology groups are functors $H^ p(K, - ) : \textit{Ab}(\mathcal{C}) \to \textit{Ab}$ since $\textit{Mod}(\mathcal{O}) = \textit{Ab}(\mathcal{C})$ in this case.
We can translate some of our already proven results using this language.
Proof. We may replace $K$ by its sheafification and assume $K$ is a sheaf. Note that both $H^ p(K, \mathcal{F})$ and $H^ p(K, \mathcal{F}_{ab})$ depend only on the topos, not on the underlying site. Hence by Sites, Lemma 7.29.5 we may replace $\mathcal{C}$ by a “larger” site such that $K = h_ U$ for some object $U$ of $\mathcal{C}$. In this case the result follows from Lemma 21.12.4. $\square$
Lemma 21.13.2. Let $\mathcal{C}$ be a site. Let $K' \to K$ be a map of presheaves of sets on $\mathcal{C}$ whose sheafification is surjective. Set $K'_ p = K' \times _ K \ldots \times _ K K'$ ($p + 1$-factors). For every abelian sheaf $\mathcal{F}$ there is a spectral sequence with $E_1^{p, q} = H^ q(K'_ p, \mathcal{F})$ converging to $H^{p + q}(K, \mathcal{F})$.
Proof. Since sheafification is exact, we see that $(K_ p')^\# $ is equal to $(K')^\# \times _{K^\# } \ldots \times _{K^\# } (K')^\# $ ($p + 1$-factors). Thus we may replace $K$ and $K'$ by their sheafifications and assume $K \to K'$ is a surjective map of sheaves. After replacing $\mathcal{C}$ by a “larger” site as in Sites, Lemma 7.29.5 we may assume that $K, K'$ are objects of $\mathcal{C}$ and that $\mathcal{U} = \{ K' \to K\} $ is a covering. Then we have the Čech to cohomology spectral sequence of Lemma 21.10.6 whose $E_1$ page is as indicated in the statement of the lemma. $\square$
Lemma 21.13.3. Let $\mathcal{C}$ be a site. Let $K$ be a sheaf of sets on $\mathcal{C}$. Consider the morphism of topoi $j : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, see Sites, Lemma 7.30.3. Then $j^{-1}$ preserves injectives and $H^ p(K, \mathcal{F}) = H^ p(\mathcal{C}/K, j^{-1}\mathcal{F})$ for any abelian sheaf $\mathcal{F}$ on $\mathcal{C}$.
Proof. By Sites, Lemmas 7.30.1 and 7.30.3 the morphism of topoi $j$ is equivalent to a localization. Hence this follows from Lemma 21.7.1. $\square$
Keeping in mind Lemma 21.13.1 we see that the following definition is the “correct one” also for sheaves of modules on ringed sites.
Definition 21.13.4. Let $\mathcal{C}$ be a site. We say an abelian sheaf $\mathcal{F}$ is totally acyclic1 if for every sheaf of sets $K$ we have $H^ p(K, \mathcal{F}) = 0$ for all $p \geq 1$.
It is clear that being totally acyclic is an intrinsic property, i.e., preserved under equivalences of topoi. A totally acyclic sheaf has vanishing higher cohomology on all objects of the site, but in general the condition of being totally acyclic is strictly stronger. Here is a characterization of totally acyclic sheaves which is sometimes useful.
Lemma 21.13.5. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be an abelian sheaf. If
$H^ p(U, \mathcal{F}) = 0$ for $p > 0$ and $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and
for every surjection $K' \to K$ of sheaves of sets the extended Čech complex
is exact,
\[ 0 \to H^0(K, \mathcal{F}) \to H^0(K', \mathcal{F}) \to H^0(K' \times _ K K', \mathcal{F}) \to \ldots \]
then $\mathcal{F}$ is totally acyclic (and the converse holds too).
Proof. By assumption (1) we have $H^ p(h_ U^\# , g^{-1}\mathcal{I}) = 0$ for all $p > 0$ and all objects $U$ of $\mathcal{C}$. Note that if $K = \coprod K_ i$ is a coproduct of sheaves of sets on $\mathcal{C}$ then $H^ p(K, g^{-1}\mathcal{I}) = \prod H^ p(K_ i, g^{-1}\mathcal{I})$. For any sheaf of sets $K$ there exists a surjection
\[ K' = \coprod h_{U_ i}^\# \longrightarrow K \]
see Sites, Lemma 7.12.5. Thus we conclude that: (*) for every sheaf of sets $K$ there exists a surjection $K' \to K$ of sheaves of sets such that $H^ p(K', \mathcal{F}) = 0$ for $p > 0$. We claim that (*) and condition (2) imply that $\mathcal{F}$ is totally acyclic. Note that conditions (*) and (2) only depend on $\mathcal{F}$ as an object of the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and not on the underlying site. (We will not use property (1) in the rest of the proof.)
We are going to prove by induction on $n \geq 0$ that (*) and (2) imply the following induction hypothesis $IH_ n$: $H^ p(K, \mathcal{F}) = 0$ for all $0 < p \leq n$ and all sheaves of sets $K$. Note that $IH_0$ holds. Assume $IH_ n$. Pick a sheaf of sets $K$. Pick a surjection $K' \to K$ such that $H^ p(K', \mathcal{F}) = 0$ for all $p > 0$. We have a spectral sequence with
\[ E_1^{p, q} = H^ q(K'_ p, \mathcal{F}) \]
covering to $H^{p + q}(K, \mathcal{F})$, see Lemma 21.13.2. By $IH_ n$ we see that $E_1^{p, q} = 0$ for $0 < q \leq n$ and by assumption (2) we see that $E_2^{p, 0} = 0$ for $p > 0$. Finally, we have $E_1^{0, q} = 0$ for $q > 0$ because $H^ q(K', \mathcal{F}) = 0$ by choice of $K'$. Hence we conclude that $H^{n + 1}(K, \mathcal{F}) = 0$ because all the terms $E_2^{p, q}$ with $p + q = n + 1$ are zero. $\square$
[1] Although this terminology is is used in [Vbis, Proposition 1.3.10, SGA4] this is probably nonstandard notation. In [V, Definition 4.1, SGA4] this property is dubbed “flasque”, but we cannot use this because it would clash with our definition of flasque sheaves on topological spaces. Please email stacks.project@gmail.com if you have a better suggestion.
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