The Stacks project

Lemma 20.19.1. Let $X$ be a ringed space. Assume that the underlying topological space of $X$ has the following properties:

  1. there exists a basis of quasi-compact open subsets, and

  2. the intersection of any two quasi-compact opens is quasi-compact.

Then for any directed system $(\mathcal{F}_ i, \varphi _{ii'})$ of sheaves of $\mathcal{O}_ X$-modules and for any quasi-compact open $U \subset X$ the canonical map

\[ \mathop{\mathrm{colim}}\nolimits _ i H^ q(U, \mathcal{F}_ i) \longrightarrow H^ q(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i) \]

is an isomorphism for every $q \geq 0$.

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 $i = 0$ 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

\[ 0 \to \mathcal{F}_ i \to \mathcal{I}_ i \to \mathcal{Q}_ i \to 0. \]

We claim that the sequence

\[ 0 \to \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i \to \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i \to \mathop{\mathrm{colim}}\nolimits _ i \mathcal{Q}_ i \to 0. \]

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

\[ \xymatrix{ \mathop{\mathrm{colim}}\nolimits _ i H^{q_0}(U, \mathcal{I}_ i) \ar[d] \ar[r] & \mathop{\mathrm{colim}}\nolimits _ i H^{q_0}(U, \mathcal{Q}_ i) \ar[d] \ar[r] & \mathop{\mathrm{colim}}\nolimits _ i H^{q_0 + 1}(U, \mathcal{F}_ i) \ar[d] \ar[r] & 0 \ar[d] \\ H^{q_0}(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) \ar[r] & H^{q_0}(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{Q}_ i) \ar[r] & H^{q_0 + 1}(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i) \ar[r] & 0 } \]

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

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) = \mathop{\mathrm{colim}}\nolimits _ i \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}_ i) \]

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. $\square$


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