The Stacks project

Proof. Set $\mathcal{D} = D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$. In both cases the complex $K = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (P, E)$ is an object of $\mathcal{D}$. See Lemma 75.13.10 for quasi-coherence. It is clear that $K$ is supported on $T$ as formation of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ commutes with restriction to opens. The map $\alpha $ defines an element of $H^0(K) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(\mathcal{O}_ X[0], K)$. Then it suffices to prove the result for the map $\alpha : \mathcal{O}_ X[0] \to K$.

Let $E \in \mathcal{D}$ be a perfect generator, see Lemma 75.15.6. Write

as in Derived Categories, Lemma 13.37.3 using the generator $E$. Since the functor $\mathcal{D} \to D(\mathcal{O}_ X)$ commutes with direct sums, we see that $K = \text{hocolim} K_ n$ holds in $D(\mathcal{O}_ X)$. Since $\mathcal{O}_ X$ is a compact object of $D(\mathcal{O}_ X)$ we find an $n$ and a morphism $\alpha _ n : \mathcal{O}_ X \to K_ n$ which gives rise to $\alpha $, see Derived Categories, Lemma 13.33.9. By Derived Categories, Lemma 13.37.4 applied to the morphism $\mathcal{O}_ X[0] \to K_ n$ in the ambient category $D(\mathcal{O}_ X)$ we see that $\alpha _ n$ factors as $\mathcal{O}_ X[0] \to Q \to K_ n$ where $Q$ is an object of $\langle E \rangle $. We conclude that $Q$ is a perfect complex supported on $T$.

Choose a distinguished triangle

By construction $I$ is perfect, the map $I \to \mathcal{O}_ X[0]$ restricts to an isomorphism over $U$, and the composition $I \to K$ is zero as $\alpha $ factors through $Q$. This proves the lemma. $\square$