The Stacks project

Lemma 36.17.5. Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \subset X$ be a closed subset such that $U = X \setminus T$ is quasi-compact. Let $\alpha : P \to E$ be a morphism of $D_\mathit{QCoh}(\mathcal{O}_ X)$ with either

  1. $P$ is perfect and $E$ supported on $T$, or

  2. $P$ pseudo-coherent, $E$ supported on $T$, and $E$ bounded below.

Then there exists a perfect complex of $\mathcal{O}_ X$-modules $I$ and a map $I \to \mathcal{O}_ X[0]$ such that $I \otimes ^\mathbf {L} P \to E$ is zero and such that $I|_ U \to \mathcal{O}_ U[0]$ is an isomorphism.

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 36.10.8 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 36.15.4. Write

\[ K = \text{hocolim} K_ n \]

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

\[ I \to \mathcal{O}_ X[0] \to Q \to I[1] \]

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. $\square$

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