Lemma 36.6.2. Let $X$ be a scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is a retrocompact open of $X$. Let $i : T \to X$ be the inclusion.

1. For $E$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ we have $i_*R\mathcal{H}_ T(E)$ in $D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$.

2. The functor $i_* \circ R\mathcal{H}_ T : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$ is right adjoint to the inclusion functor $D_{\mathit{QCoh}, T}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ X)$.

Proof. Set $U = X \setminus T$ and denote $j : U \to X$ the inclusion. By Cohomology, Lemma 20.34.6 there is a distinguished triangle

$i_*R\mathcal{H}_ T(E) \to E \to Rj_*(E|_ U) \to i_*R\mathcal{H}_ Z(E)[1]$

in $D(\mathcal{O}_ X)$. By Lemma 36.4.1 the complex $Rj_*(E|_ U)$ has quasi-coherent cohomology sheaves (this is where we use that $U$ is retrocompact in $X$). Thus we see that (1) is true. Part (2) follows from this and the adjointness of functors in Cohomology, Lemma 20.34.2. $\square$

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