Lemma 56.6.4. Let $X$ be a quasi-compact and quasi-separated scheme. Let $Z \subset X$ be a closed subset such that $U = X \setminus Z$ is quasi-compact. Let $\mathcal{A} \subset \mathit{QCoh}(\mathcal{O}_ X)$ be the full subcategory whose objects are the quasi-coherent modules supported on $Z$. Then the restriction functor $\mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ U)$ induces an equivalence $\mathit{QCoh}(\mathcal{O}_ X)/\mathcal{A} \cong \mathit{QCoh}(\mathcal{O}_ U)$.

Proof. By the universal property of the quotient construction (Homology, Lemma 12.10.6) we certainly obtain an induced functor $\mathit{QCoh}(\mathcal{O}_ X)/\mathcal{A} \cong \mathit{QCoh}(\mathcal{O}_ U)$. Denote $j : U \to X$ the inclusion morphism. Since $j$ is quasi-compact and quasi-separated we obtain a functor $j_* : \mathit{QCoh}(\mathcal{O}_ U) \to \mathit{QCoh}(\mathcal{O}_ X)$. The reader shows that this defines a quasi-inverse; details omitted. $\square$

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