Lemma 75.14.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ which is representable by an affine scheme. Then approximation holds for every triple $(T, E, m)$ as in Definition 75.14.1 such that there exists an integer $r \geq 0$ with
$E$ is $m$-pseudo-coherent,
$H^ i(E)$ is supported on $T$ for $i \geq m - r + 1$,
$X \setminus T$ is the union of $r$ affine opens.
In particular, approximation by perfect complexes holds for affine schemes.
Proof. Let $X_0$ be an affine scheme representing $X$. Let $T_0 \subset X_0$ by the closed subset corresponding to $T$. Let $\epsilon : X_{\acute{e}tale}\to X_{0, Zar}$ be the morphism (75.4.0.1). We may write $E = \epsilon ^*E_0$ for some object $E_0$ of $D_\mathit{QCoh}(\mathcal{O}_{X_0})$, see Lemma 75.4.2. Then $E_0$ is $m$-pseudo-coherent, see Lemma 75.13.2. Comparing stalks of cohomology sheaves (see proof of Lemma 75.4.1) we see that $H^ i(E_0)$ is supported on $T_0$ for $i \geq m - r + 1$. By Derived Categories of Schemes, Lemma 36.14.4 there exists an approximation $P_0 \to E_0$ of $(T_0, E_0, m)$. By Lemma 75.13.5 we see that $P = \epsilon ^*P_0$ is a perfect object of $D(\mathcal{O}_ X)$. Pulling back we obtain an approximation $P = \epsilon ^*P_0 \to \epsilon ^*E_0 = E$ as desired. $\square$
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