Lemma 21.13.5. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be an abelian sheaf. If

1. $H^ p(U, \mathcal{F}) = 0$ for $p > 0$ and $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and

2. for every surjection $K' \to K$ of sheaves of sets the extended Čech complex

$0 \to H^0(K, \mathcal{F}) \to H^0(K', \mathcal{F}) \to H^0(K' \times _ K K', \mathcal{F}) \to \ldots$

is exact,

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. $\square$

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