Lemma 36.14.5. Let $X$ be a scheme. Let $X = U \cup V$ be an open covering with $U$ quasi-compact, $V$ affine, and $U \cap V$ quasi-compact. If approximation by perfect complexes holds on $U$, then approximation holds on $X$.
Proof. Let $T \subset X$ be a closed subset with $X \setminus T$ retro-compact in $X$. Let $r_ U$ be the integer of Definition 36.14.2 adapted to the pair $(U, T \cap U)$. Set $T' = T \setminus U$. Note that $T' \subset V$ and that $V \setminus T' = (X \setminus T) \cap U \cap V$ is quasi-compact by our assumption on $T$. Let $r'$ be the number of affines needed to cover $V \setminus T'$. We claim that $r = \max (r_ U, r')$ works for the pair $(X, T)$.
To see this choose a triple $(T, E, m)$ such that $E$ is $(m - r)$-pseudo-coherent and $H^ i(E)$ is supported on $T$ for $i \geq m - r$. Let $t$ be the largest integer such that $H^ t(E)|_ U$ is nonzero. (Such an integer exists as $U$ is quasi-compact and $E|_ U$ is $(m - r)$-pseudo-coherent.) We will prove that $E$ can be approximated by induction on $t$.
Base case: $t \leq m - r'$. This means that $H^ i(E)$ is supported on $T'$ for $i \geq m - r'$. Hence Lemma 36.14.4 guarantees the existence of an approximation $P \to E|_ V$ of $(T', E|_ V, m)$ on $V$. Applying Lemma 36.14.3 we see that $(T', E, m)$ can be approximated. Such an approximation is also an approximation of $(T, E, m)$.
Induction step. Choose an approximation $P \to E|_ U$ of $(T \cap U, E|_ U, m)$. This in particular gives a surjection $H^ t(P) \to H^ t(E|_ U)$. By Lemma 36.13.9 we can choose a perfect object $Q$ in $D(\mathcal{O}_ V)$ supported on $T \cap V$ and an isomorphism $Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V}$. By Lemma 36.13.6 we can replace $Q$ by $Q \otimes ^\mathbf {L} I$ and assume that the map
\[ Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V} \longrightarrow P|_{U \cap V} \longrightarrow E|_{U \cap V} \]
lifts to $Q \to E|_ V$. By Cohomology, Lemma 20.45.1 we find an morphism $a : R \to E$ of $D(\mathcal{O}_ X)$ such that $a|_ U$ is isomorphic to $P \oplus P[1] \to E|_ U$ and $a|_ V$ isomorphic to $Q \to E|_ V$. Thus $R$ is perfect and supported on $T$ and the map $H^ t(R) \to H^ t(E)$ is surjective on restriction to $U$. Choose a distinguished triangle
\[ R \to E \to E' \to R[1] \]
Then $E'$ is $(m - r)$-pseudo-coherent (Cohomology, Lemma 20.47.4), $H^ i(E')|_ U = 0$ for $i \geq t$, and $H^ i(E')$ is supported on $T$ for $i \geq m - r$. By induction we find an approximation $R' \to E'$ of $(T, E', m)$. Fit the composition $R' \to E' \to R[1]$ into a distinguished triangle $R \to R'' \to R' \to R[1]$ and extend the morphisms $R' \to E'$ and $R[1] \to R[1]$ into a morphism of distinguished triangles
\[ \xymatrix{ R \ar[r] \ar[d] & R'' \ar[d] \ar[r] & R' \ar[d] \ar[r] & R[1] \ar[d] \\ R \ar[r] & E \ar[r] & E' \ar[r] & R[1] } \]
using TR3. Then $R''$ is a perfect complex (Cohomology, Lemma 20.49.7) supported on $T$. An easy diagram chase shows that $R'' \to E$ is the desired approximation. $\square$
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