The Stacks project

Lemma 36.33.3. Let $X$ be a quasi-compact and quasi-separated scheme such that the sets of sections of $\mathcal{O}_ X$ over affine opens are countable. Let $K$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. The following are equivalent

  1. $K = \text{hocolim} E_ n$ with $E_ n$ a perfect object of $D(\mathcal{O}_ X)$, and

  2. the cohomology sheaves $H^ i(K)$ have countable sets of sections over affine opens.

Proof. If (1) is true, then (2) is true because homotopy colimits commutes with taking cohomology sheaves (by Derived Categories, Lemma 13.33.8) and because a perfect complex is locally isomorphic to a finite complex of finite free $\mathcal{O}_ X$-modules and therefore satisfies (2) by assumption on $X$.

Assume (2). Choose a K-injective complex $\mathcal{K}^\bullet $ representing $K$. Choose a perfect generator $E$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ and represent it by a K-injective complex $\mathcal{I}^\bullet $. According to Theorem 36.18.2 and its proof there is an equivalence of triangulated categories $F : D_\mathit{QCoh}(\mathcal{O}_ X) \to D(A, \text{d})$ where $(A, \text{d})$ is the differential graded algebra

\[ (A, \text{d}) = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O}_ X)} (\mathcal{I}^\bullet , \mathcal{I}^\bullet ) \]

which maps $K$ to the differential graded module

\[ M = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O}_ X)} (\mathcal{I}^\bullet , \mathcal{K}^\bullet ) \]

Note that $H^ i(A) = \mathop{\mathrm{Ext}}\nolimits ^ i(E, E)$ and $H^ i(M) = \mathop{\mathrm{Ext}}\nolimits ^ i(E, K)$. Moreover, since $F$ is an equivalence it and its quasi-inverse commute with homotopy colimits. Therefore, it suffices to write $M$ as a homotopy colimit of compact objects of $D(A, \text{d})$. By Differential Graded Algebra, Lemma 22.38.3 it suffices show that $\mathop{\mathrm{Ext}}\nolimits ^ i(E, E)$ and $\mathop{\mathrm{Ext}}\nolimits ^ i(E, K)$ are countable for each $i$. This follows from Lemma 36.33.2. $\square$


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