The Stacks project

Proof. It is important in this proof to argue for all quasi-compact opens $U \subset X$ at the same time. The result is true for $q = 0$ and any quasi-compact open $U \subset X$ by Sheaves, Lemma 6.29.1 (combined with Topology, Lemma 5.27.1). Assume that we have proved the result for all $q \leq q_0$ and let us prove the result for $q = q_0 + 1$.

By our conventions on directed systems the index set $I$ is directed, and any system of $\mathcal{O}_ X$-modules $(\mathcal{F}_ i, \varphi _{ii'})$ over $I$ is directed. By Injectives, Lemma 19.5.1 the category of $\mathcal{O}_ X$-modules has functorial injective embeddings. Thus for any system $(\mathcal{F}_ i, \varphi _{ii'})$ there exists a system $(\mathcal{I}_ i, \varphi _{ii'})$ with each $\mathcal{I}_ i$ an injective $\mathcal{O}_ X$-module and a morphism of systems given by injective $\mathcal{O}_ X$-module maps $\mathcal{F}_ i \to \mathcal{I}_ i$. Denote $\mathcal{Q}_ i$ the cokernel so that we have short exact sequences

We claim that the sequence

is also a short exact sequence of $\mathcal{O}_ X$-modules. We may check this on stalks. By Sheaves, Sections 6.28 and 6.29 taking stalks commutes with colimits. Since a directed colimit of short exact sequences of abelian groups is short exact (see Algebra, Lemma 10.8.8) we deduce the result. We claim that $H^ q(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) = 0$ for all quasi-compact open $U \subset X$ and all $q \geq 1$. Accepting this claim for the moment consider the diagram

The zero at the lower right corner comes from the claim and the zero at the upper right corner comes from the fact that the sheaves $\mathcal{I}_ i$ are injective. The top row is exact by an application of Algebra, Lemma 10.8.8. Hence by the snake lemma we deduce the result for $q = q_0 + 1$.

It remains to show that the claim is true. We will use Lemma 20.11.9. Let $\mathcal{B}$ be the collection of all quasi-compact open subsets of $X$. This is a basis for the topology on $X$ by assumption. Let $\text{Cov}$ be the collection of finite open coverings $\mathcal{U} : U = \bigcup _{j = 1, \ldots , m} U_ j$ with each of $U$, $U_ j$ quasi-compact open in $X$. By the result for $q = 0$ we see that for $\mathcal{U} \in \text{Cov}$ we have

because all the multiple intersections $U_{j_0 \ldots j_ p}$ are quasi-compact. By Lemma 20.11.1 each of the complexes in the colimit of Čech complexes is acyclic in degree $\geq 1$. Hence by Algebra, Lemma 10.8.8 we see that also the Čech complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i)$ is acyclic in degrees $\geq 1$. In other words we see that $\check{H}^ p(\mathcal{U}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) = 0$ for all $p \geq 1$. Thus the assumptions of Lemma 20.11.9 are satisfied and the claim follows. $\square$