Lemma 47.10.2. Let $A$ be a Noetherian ring and let $I = (f_1, \ldots , f_ r)$ be an ideal of $A$. Set $Z = V(I) \subset \mathop{\mathrm{Spec}}(A)$. There are canonical isomorphisms

\[ R\Gamma _ I(A) \to (A \to \prod \nolimits _{i_0} A_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_ r}) \to R\Gamma _ Z(A) \]

in $D(A)$. If $M$ is an $A$-module, then we have similarly

\[ R\Gamma _ I(M) \cong (M \to \prod \nolimits _{i_0} M_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} M_{f_{i_0}f_{i_1}} \to \ldots \to M_{f_1\ldots f_ r}) \cong R\Gamma _ Z(M) \]

in $D(A)$.

## Comments (3)

Comment #3596 by Kestutis Cesnavicius on

Comment #3598 by Kestutis Cesnavicius on

Comment #3713 by Johan on