Lemma 21.43.6 (0GYX)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.43: Modules on a category
Lemma 21.43.6 (cite)

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Lemma 21.43.6. In the situation above, there exists a cardinal $\kappa$ with the following property: given a complex $\mathcal{F}^\bullet$ of $\mathcal{O}$ -modules and subsets $\Omega ^ i_ U \subset \mathcal{F}^ i(U)$ there exists a subcomplex $\mathcal{H}^\bullet \subset \mathcal{F}^\bullet$ with $\Omega ^ i_ U \subset \mathcal{H}^ i(U)$ and $|\mathcal{H}^\bullet | \leq \max (\kappa , |\bigcup \Omega ^ i_ U|)$ .

Proof. Define $\mathcal{H}^ i(U)$ to be the $\mathcal{O}(U)$ -submodule of $\mathcal{F}^ i(U)$ generated by the images of $\Omega ^ i_ V$ and $\text{d}(\Omega ^{i - 1}_ U)$ by restriction along any morphism $f : U \to V$ . The cardinality of $\mathcal{H}^ i(U)$ is bounded by the maximum of $\aleph _0$ , the cardinality of the $\mathcal{O}(U)$ , the cardinality of $\text{Arrows}(\mathcal{C})$ , and $|\bigcup \Omega ^ i_ U|$ . Details omitted. $\square$

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