The Stacks project

Proof. Say $X = \mathop{\mathrm{Spec}}(A)$. Write $T = V(f_1, \ldots , f_ r)$. (The case $r = 0$, i.e., $T = X$ follows immediately from Lemma 36.10.2 and the definitions.) Let $(T, E, m)$ be a triple as in the lemma. Let $t$ be the largest integer such that $H^ t(E)$ is nonzero. We will proceed by induction on $t$. The base case is $t < m$; in this case the result is trivial. Now suppose that $t \geq m$. By Cohomology, Lemma 20.47.9 the sheaf $H^ t(E)$ is of finite type. Since it is quasi-coherent it is generated by finitely many sections (Properties, Lemma 28.16.1). For every $s \in \Gamma (X, H^ t(E)) = H^ t(X, E)$ (see proof of Lemma 36.3.5) we can find an $e > 0$ and a morphism $K_ e[-t] \to E$ such that $s$ is in the image of $H^0(K_ e) = H^ t(K_ e[-t]) \to H^ t(E)$, see Lemma 36.9.6. Taking a finite direct sum of these maps we obtain a map $P \to E$ where $P$ is a perfect complex supported on $T$, where $H^ i(P) = 0$ for $i > t$, and where $H^ t(P) \to E$ is surjective. Choose a distinguished triangle

Then $E'$ is $m$-pseudo-coherent (Cohomology, Lemma 20.47.4), $H^ i(E') = 0$ for $i \geq t$, and $H^ i(E')$ is supported on $T$ for $i \geq m - r + 1$. By induction we find an approximation $P' \to E'$ of $(T, E', m)$. Fit the composition $P' \to E' \to P[1]$ into a distinguished triangle $P \to P'' \to P' \to P[1]$ and extend the morphisms $P' \to E'$ and $P[1] \to P[1]$ into a morphism of distinguished triangles

using TR3. Then $P''$ is a perfect complex (Cohomology, Lemma 20.49.7) supported on $T$. An easy diagram chase shows that $P'' \to E$ is the desired approximation. $\square$