The Stacks project

Lemma 73.27.1. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $K$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$ such that the cohomology sheaves $H^ i(K)$ have countable sets of sections over affine schemes ├ętale over $X$. Then for any quasi-compact and quasi-separated ├ętale morphism $U \to X$ and any perfect object $E$ in $D(\mathcal{O}_ X)$ the sets

\[ H^ i(U, K \otimes ^\mathbf {L} E),\quad \mathop{\mathrm{Ext}}\nolimits ^ i(E|_ U, K|_ U) \]

are countable.

Proof. Using Cohomology on Sites, Lemma 21.46.4 we see that it suffices to prove the result for the groups $H^ i(U, K \otimes ^\mathbf {L} E)$. We will use the induction principle to prove the lemma, see Lemma 73.9.3.

When $U = \mathop{\mathrm{Spec}}(A)$ is affine the result follows from the case of schemes, see Derived Categories of Schemes, Lemma 36.33.2.

To finish the proof it suffices to show: if $(U \subset W, V \to W)$ is an elementary distinguished triangle and the result holds for $U$, $V$, and $U \times _ W V$, then the result holds for $W$. This is an immediate consquence of the Mayer-Vietoris sequence, see Lemma 73.10.5. $\square$

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