Lemma 47.10.3. If $A \to B$ is a homomorphism of Noetherian rings and $I \subset A$ is an ideal, then in $D(B)$ we have

\[ R\Gamma _ I(A) \otimes _ A^\mathbf {L} B = R\Gamma _ Z(A) \otimes _ A^\mathbf {L} B = R\Gamma _ Y(B) = R\Gamma _{IB}(B) \]

where $Y = V(IB) \subset \mathop{\mathrm{Spec}}(B)$.

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