The Stacks project

Lemma 48.12.2. Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \to Y$ be a morphism of schemes which is proper, flat, and of finite presentation. Let $a$ be the right adjoint for $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ of Lemma 48.3.1. Then

1. for every closed $T \subset Y$ if $Q \in D_\mathit{QCoh}(Y)$ is supported on $T$, then $a(Q)$ is supported on $f^{-1}(T)$,

2. for every open $V \subset Y$ and any $K \in D_\mathit{QCoh}(\mathcal{O}_ Y)$ the map (48.4.1.1) is an isomorphism, and