The Stacks project

Proof. Let $\kappa$ be an upper bound for the following set of cardinals:

1. $|\coprod _ V j_{U!}\mathcal{O}_ U(V)|$ for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$,

2. the cardinals $\kappa (\mathcal{O}(V) \to \mathcal{O}(U))$ found in More on Algebra, Lemma 15.102.5 for all morphisms $U \to V$ in $\mathcal{C}$,

3. the cardinal found in Lemma 21.43.7.

We claim that for any complex $\mathcal{F}^\bullet$ representing an object of $\mathit{QC}(\mathcal{O})$ and any subcomplex $\mathcal{S}^\bullet \subset \mathcal{F}^\bullet$ with $|\mathcal{S}^\bullet | \leq \kappa$ there exists a subcomplex $\mathcal{H}^\bullet$ of $\mathcal{F}^\bullet$ containing $\mathcal{S}^\bullet$ such that $\mathcal{H}^\bullet$ represents an object of $\mathit{QC}(\mathcal{O})$ and such that $|\mathcal{H}^\bullet | \leq \kappa$. In the next two paragraphs we show that the claim implies the lemma.

As in (1) let $K$ be a nonzero object of $\mathit{QC}(\mathcal{O})$. Say $K$ is represented by the complex of $\mathcal{O}$-modules $\mathcal{F}^\bullet$. Then $H^ i(\mathcal{F}^\bullet )$ is nonzero for some $i$. Hence there exists an object $U$ of $\mathcal{C}$ and a section $s \in \mathcal{F}^ i(U)$ with $d(s) = 0$ which determines a nonzero section of $H^ i(\mathcal{F}^\bullet )$ over $U$. Then the image of $s : j_{U!}\mathcal{O}_ U[-i] \to \mathcal{F}^\bullet$ is a subcomplex $\mathcal{S}^\bullet \subset \mathcal{F}^\bullet$ with $|\mathcal{S}^\bullet | \leq \kappa$. Applying the claim we get $\mathcal{H}^\bullet \to \mathcal{F}^\bullet$ in $\mathit{QC}(\mathcal{O})$ nonzero with $|\mathcal{H}^\bullet | \leq \kappa$. Thus (1) holds.

Let $\alpha : E \to \bigoplus K_ n$ be as in (2). Choose any complexes $\mathcal{K}_ n^\bullet$ representing $K_ n$. Then $\bigoplus \mathcal{K}_ n^\bullet$ represents $\bigoplus K_ n$. By the construction of the derived category we can represent $E$ by a complex $\mathcal{E}^\bullet$ such that $\alpha$ is represented by a morphism $a : \mathcal{E}^\bullet \to \bigoplus \mathcal{K}_ n^\bullet$ of complexes. By Lemma 21.43.7 and our choice of $\kappa$ above we may assume $|\mathcal{E}^\bullet | \leq \kappa$. By the claim we get subcomplexes $\mathcal{E}_ n^\bullet \subset \mathcal{K}_ n^\bullet$ representing objects $E_ n$ of $\mathit{QC}(\mathcal{O})$ with $|E_ n| \leq \kappa$ containing the image of $a_ n : \mathcal{E}^\bullet \to \mathcal{K}_ n^\bullet$ as desired.

Proof of the claim. Let $\mathcal{F}^\bullet$ be a complex representing an object of $\mathit{QC}(\mathcal{O})$ and let $\mathcal{S}^\bullet \subset \mathcal{F}^\bullet$ be a subcomplex of size $\leq \kappa$. We are going to inductively construct subcomplexes

of size $\leq \kappa$ such that for every morphism $f : U \to V$ of $\mathcal{C}$ and every $i \in \mathbf{Z}$

1. the kernel of the arrow $H^ i(\mathcal{S}_ n^\bullet (V) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U)) \to H^ i(\mathcal{S}_ n^\bullet (U))$ maps to zero in $H^ i(\mathcal{S}_{n + 1}^\bullet (V) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U))$,

2. the image of the arrow $H^ i(\mathcal{S}_ n^\bullet (U)) \to H^ i(\mathcal{S}_{n + 1}^\bullet (U))$ is contained in the image of $H^ i(\mathcal{S}_{n + 1}^\bullet (V) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U)) \to H^ i(\mathcal{S}_{n + 1}^\bullet (U))$,

Once this is done we can set $\mathcal{H}^\bullet = \bigcup \mathcal{S}_ n^\bullet$. Namely, since derived tensor product and taking cohomology of complexes of modules over rings commute with filtered colimits, the conditions (1) and (2) together will guarantee that

is an isomorphism on cohomology in all degrees and hence an isomorphism in $D(\mathcal{O}(U))$ for all $f : U \to V$ in $\mathcal{C}$. Hence $\mathcal{H}^\bullet$ represents an object of $\mathit{QC}(\mathcal{O})$ as desired.

Construction of $\mathcal{S}_{n + 1}$ given $\mathcal{S}_ n$. For every morphism $f : U \to V$ of $\mathcal{C}$ we consider the commutative diagram

This is a diagram as in More on Algebra, Lemma 15.102.5 for the ring map $\mathcal{O}(V) \to \mathcal{O}(U)$, i.e., the bottom row induces an isomorphism

in $D(\mathcal{O}(U))$. Thus we may choose subcomplexes

as in More on Algebra, Lemma 15.102.5 and in particular we see that $|N^ i_ f|, |M^ i_ f| \leq \kappa$. Next, we apply Lemma 21.43.6 using the subsets

to find a subcomplex

with containing those subsets and such that $|\mathcal{S}_{n + 1}^\bullet | \leq \kappa$. Conditions (1) and (2) hold because the corresponding statements hold for $\mathcal{S}_ n^\bullet (V) \subset M^\bullet _ f$ and $\mathcal{S}_ n^\bullet (U) \subset N^\bullet _ f$ by the construction in More on Algebra, Lemma 15.102.5. Thus the proof is complete. $\square$