Lemma 36.14.3. Let $X$ be a scheme. Let $U \subset X$ be an open subscheme. Let $(T, E, m)$ be a triple as in Definition 36.14.1. If
$T \subset U$,
approximation holds for $(T, E|_ U, m)$, and
the sheaves $H^ i(E)$ for $i \geq m$ are supported on $T$,
then approximation holds for $(T, E, m)$.
Proof. Let $j : U \to X$ be the inclusion morphism. If $P \to E|_ U$ is an approximation of the triple $(T, E|_ U, m)$ over $U$, then $j_!P = Rj_*P \to j_!(E|_ U) \to E$ is an approximation of $(T, E, m)$ over $X$. See Cohomology, Lemmas 20.33.6 and 20.49.10. $\square$
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