The Stacks project

Lemma 51.5.3. Let $A$ be a Noetherian ring. Let $T \subset \mathop{\mathrm{Spec}}(A)$ be a subset stable under specialization. For any object $K$ of $D(A)$ we have

\[ H^ i(RH^0_ T(K)) = \mathop{\mathrm{colim}}\nolimits _{Z \subset T\text{ closed}} H^ i_ Z(K) \]

Proof. Let $J^\bullet $ be a K-injective complex representing $K$. By definition $RH^0_ T$ is represented by the complex

\[ H^0_ T(J^\bullet ) = \mathop{\mathrm{colim}}\nolimits H^0_ Z(J^\bullet ) \]

where the equality follows from our definition of $H^0_ T$. Since filtered colimits are exact the cohomology of this complex in degree $i$ is $\mathop{\mathrm{colim}}\nolimits H^ i(H^0_ Z(J^\bullet )) = \mathop{\mathrm{colim}}\nolimits H^ i_ Z(K)$ as desired. $\square$

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