Lemma 51.5.2. Let $A$ be a Noetherian ring. Let $T \subset \mathop{\mathrm{Spec}}(A)$ be a subset stable under specialization. The functor $RH^0_ T$ is the right adjoint to the functor $D(\text{Mod}_{A, T}) \to D(A)$.

Proof. This follows from the fact that the functor $H^0_ T(-)$ is the right adjoint to the inclusion functor $\text{Mod}_{A, T} \to \text{Mod}_ A$, see Derived Categories, Lemma 13.30.3. $\square$

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