The Stacks project

Proof. Being perfect is local on $X_{\acute{e}tale}$. Thus it suffices to check that $Rj_*E$ is perfect when restricted to $U$ and $V$. We have $Rj_*E|_ V = E$ by Lemma 75.10.7 which is perfect. We have $Rj_*E|_ U = 0$ because $E|_{V \setminus j^{-1}(T)} = 0$ (use Lemma 75.3.1). $\square$

Proof. Let $P \to E|_ V$ be an approximation of the triple $(j^{-1}T, E|_ V, m)$ over $V$. Then $Rj_*P$ is a perfect object of $D(\mathcal{O}_ X)$ by Lemma 75.14.3. On the other hand, $Rj_*P = j_!P$ by Lemma 75.10.7. We see that $j_!P$ is supported on $T$ for example by (75.10.0.2). Hence we obtain an approximation $Rj_*P = j_!P \to j_!(E|_ V) \to E$. $\square$

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. $\square$

Proof. Let $T \subset |X|$ be a closed subset with $X \setminus T \to X$ quasi-compact. Let $r_ U$ be the integer of Definition 75.14.2 adapted to the pair $(U, T \cap |U|)$. Set $T' = T \setminus |U|$. Endow $T'$ with the induced reduced subspace structure. Since $|T'|$ is contained in $|X| \setminus |U|$ we see that $j^{-1}(T') \to T'$ is an isomorphism. Moreover, $V \setminus j^{-1}(T')$ is quasi-compact as it is the fibre product of $U \times _ X V$ with $X \setminus T$ over $X$ and we've assumed $U \times _ X V$ quasi-compact and $X \setminus T \to X$ quasi-compact. Let $r'$ be the number of affines needed to cover $V \setminus j^{-1}(T')$. 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 75.14.5 guarantees the existence of an approximation $P \to E|_ V$ of $(T', E|_ V, m)$ on $V$. Applying Lemma 75.14.4 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 $H^ t(P) \to H^ t(E|_ U)$. In the rest of the proof we will use the equivalence of Lemma 75.4.2 (and the compatibilities of Remark 75.6.3) for the representable algebraic spaces $V$ and $U \times _ X V$. We will also use the fact that $(m - r)$-pseudo-coherence, resp. perfectness on the Zariski site and étale site agree, see Lemmas 75.13.2 and 75.13.5. Thus we can use the results of Derived Categories of Schemes, Section 36.13 for the open immersion $U \times _ X V \subset V$. In this way Derived Categories of Schemes, Lemma 36.13.10 implies there exists a perfect object $Q$ in $D(\mathcal{O}_ V)$ supported on $j^{-1}(T)$ and an isomorphism $Q|_{U \times _ X V} \to (P \oplus P[1])|_{U \times _ X V}$. By Derived Categories of Schemes, Lemma 36.13.7 we can replace $Q$ by $Q \otimes ^\mathbf {L} I$ and assume that the map

lifts to $Q \to E|_ V$. By Lemma 75.10.8 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

Then $E'$ is $(m - r)$-pseudo-coherent (Cohomology on Sites, Lemma 21.45.4), $H^ i(E')|_ U = 0$ 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

using TR3. Then $R''$ is a perfect complex (Cohomology on Sites, Lemma 21.47.6) supported on $T$. An easy diagram chase shows that $R'' \to E$ is the desired approximation. $\square$

Proof. This follows from the induction principle of Lemma 75.9.3 and Lemmas 75.14.6 and 75.14.5. $\square$