Lemma 21.43.7. In the situation above, there exists a cardinal $\kappa $ with the following property: given a complex $\mathcal{F}^\bullet $ of $\mathcal{O}$-modules representing an object $K$ of $D(\mathcal{O})$ there exists a subcomplex $\mathcal{H}^\bullet \subset \mathcal{F}^\bullet $ such that $\mathcal{H}^\bullet $ represents $K$ and such that $|\mathcal{H}^\bullet | \leq \max (\kappa , |K|)$.
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
\[ \mathcal{S}^\bullet = \mathcal{S}_0^\bullet \subset \mathcal{S}_1^\bullet \subset \mathcal{S}_2^\bullet \subset \ldots \subset \mathcal{F}^\bullet \]
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
\[ \mathcal{S}_ n^ i(U) \cap \text{Im}(\text{d} : \mathcal{F}^{i - 1}(U)\to \mathcal{F}^ i(U)) \]
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
\[ \mathcal{S}^ i(U) \cup \Omega ^ i_ U \subset \mathcal{F}^ i(U) \]
and everything is clear. $\square$
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