Lemma 51.5.3 (0EF1)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 51: Local Cohomology
Section 51.5: More general supports
Lemma 51.5.3 (cite)

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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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