The Stacks project

Lemma 73.27.2. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Assume the sets of sections of $\mathcal{O}_ X$ over affines ├ętale over $X$ 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 affines ├ętale over $X$.

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 73.17.3 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 73.27.1. $\square$


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