The Stacks project

Proof. First, for every $i$ and $U$ we choose a subset $\Omega ^ i_ U \subset \mathop{\mathrm{Ker}}(\text{d} : \mathcal{F}^ i(U) \to \mathcal{F}^{i + 1}(U))$ mapping bijectively onto $H^ i(K)(U) = H^ i(\mathcal{F}^\bullet (U))$. Hence $|\Omega ^ i_ U| \leq |K|$ as we may represent $K$ by a complex whose size is $|K|$. Applying Lemma 21.43.6 we find a subcomplex $\mathcal{S}^\bullet \subset \mathcal{F}^\bullet$ of size at most $\max (\kappa , |K|)$ containing $\Omega ^ i_ U$ and hence such that $H^ i(\mathcal{S}^\bullet ) \to H^ i(\mathcal{F}^\bullet )$ is a surjection of sheaves.

We are going to inductively construct subcomplexes

of size $\leq \max (\kappa , |K|)$ such that the kernel of $H^ i(\mathcal{S}_ n^\bullet ) \to H^ i(\mathcal{F}^\bullet )$ is the same as the kernel of $H^ i(\mathcal{S}_ n^\bullet ) \to H^ i(\mathcal{S}_{n + 1}^\bullet )$. Once this is done we can take $\mathcal{H}^\bullet = \bigcup \mathcal{S}_ n^\bullet$ as our solution.

Construction of $\mathcal{S}_{n + 1}^\bullet$ given $\mathcal{S}_ n^\bullet$. For ever $U$ and $i$ let $\Omega ^{i - 1}_ U \subset \mathcal{F}^{i - 1}(U)$ be a subset such that $\text{d} : \mathcal{F}^{i - 1}(U) \to \mathcal{F}^ i(U)$ maps $\Omega ^{i - 1}_ U$ bijectively onto

Observe that $|\Omega ^ i_ U| \leq |K|$ because $\mathcal{S}_ n^ i(U)$ is so bounded. Then we get $\mathcal{S}_{n + 1}^\bullet$ by an application of Lemma 21.43.6 to the subsets

and everything is clear. $\square$