Lemma 48.8.3. Let $f : X \to Y$ be a proper morphism of Noetherian schemes. Let $V \subset Y$ be an open such that $f^{-1}(V) \to V$ is an isomorphism. Then for $K \in D_\mathit{QCoh}^+(\mathcal{O}_ Y)$ the map (48.8.0.1) restricts to an isomorphism over $f^{-1}(V)$.

**Proof.**
By Lemma 48.4.4 the map (48.4.1.1) is an isomorphism for objects of $D_\mathit{QCoh}^+(\mathcal{O}_ Y)$. Hence Lemma 48.8.2 tells us the restriction of (48.8.0.1) for $K$ to $f^{-1}(V)$ is the map (48.8.0.1) for $K|_ V$ and $f^{-1}(V) \to V$. Thus it suffices to show that the map is an isomorphism when $f$ is the identity morphism. This is clear.
$\square$

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