Lemma 47.8.2. Let A be a ring and let I \subset A be a finitely generated ideal. For any object K of D(A) we have
R\Gamma _ I(K) = \text{hocolim}\ R\mathop{\mathrm{Hom}}\nolimits _ A(A/I^ n, K)
in D(A) and
R^ q\Gamma _ I(K) = \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Ext}}\nolimits _ A^ q(A/I^ n, K)
as modules for all q \in \mathbf{Z}.
Proof.
Let J^\bullet be a K-injective complex representing K. Then
R\Gamma _ I(K) = J^\bullet [I^\infty ] = \mathop{\mathrm{colim}}\nolimits J^\bullet [I^ n] = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ A(A/I^ n, J^\bullet )
where the first equality is the definition of R\Gamma _ I(K). By Derived Categories, Lemma 13.33.7 we obtain the first displayed equality in the statement of the lemma. The second displayed equality in the statement of the lemma then follows because H^ q(\mathop{\mathrm{Hom}}\nolimits _ A(A/I^ n, J^\bullet )) = \mathop{\mathrm{Ext}}\nolimits ^ q_ A(A/I^ n, K) and because filtered colimits are exact in the category of abelian groups.
\square
Comments (2)
Comment #6521 by Mingchen on
Comment #6578 by Johan on
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