Remark 51.5.6. Let $A$ be a Noetherian ring. Let $T \subset \mathop{\mathrm{Spec}}(A)$ be a subset stable under specialization. The upshot of the discussion above is that $R\Gamma _ T : D^+(A) \to D_ T^+(A)$ is the right adjoint to the inclusion functor $D_ T^+(A) \to D^+(A)$. If $\dim (A) < \infty$, then $R\Gamma _ T : D(A) \to D_ T(A)$ is the right adjoint to the inclusion functor $D_ T(A) \to D(A)$. In both cases we have

$H^ i_ T(K) = H^ i(R\Gamma _ T(K)) = R^ iH^0_ T(K) = \mathop{\mathrm{colim}}\nolimits _{Z \subset T\text{ closed}} H^ i_ Z(K)$

This follows by combining Lemmas 51.5.2, 51.5.3, 51.5.4, and 51.5.5.

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