Lemma 47.9.8. Let A \to B be a flat ring map and let I \subset A be a finitely generated ideal such that A/I \to B/IB is an isomorphism. For K \in D_{I^\infty \text{-torsion}}(A) and L \in D(A) the map
R\mathop{\mathrm{Hom}}\nolimits _ A(K, L) \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ B(K \otimes _ A B, L \otimes _ A B)
is a quasi-isomorphism. In particular, if M, N are A-modules and M is I-power torsion, then the canonical map
\mathop{\mathrm{Ext}}\nolimits ^ i_ A(M, N) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_ B(M \otimes _ A B, N \otimes _ A B)
is an isomorphism for all i.
Proof.
Let Z = V(I) \subset \mathop{\mathrm{Spec}}(A) and Y = V(IB) \subset \mathop{\mathrm{Spec}}(B). Since the cohomology modules of K are I power torsion, the canonical map R\Gamma _ Z(L) \to L induces an isomorphism
R\mathop{\mathrm{Hom}}\nolimits _ A(K, R\Gamma _ Z(L)) \to R\mathop{\mathrm{Hom}}\nolimits _ A(K, L)
in D(A). Similarly, the cohomology modules of K \otimes _ A B are IB power torsion and we have an isomorphism
R\mathop{\mathrm{Hom}}\nolimits _ B(K \otimes _ A B, R\Gamma _ Y(L \otimes _ A B)) \to R\mathop{\mathrm{Hom}}\nolimits _ B(K \otimes _ A B, L \otimes _ A B)
in D(B). By Lemma 47.9.3 we have R\Gamma _ Z(L) \otimes _ A B = R\Gamma _ Y(L \otimes _ A B). Hence it suffices to show that the map
R\mathop{\mathrm{Hom}}\nolimits _ A(K, R\Gamma _ Z(L)) \to R\mathop{\mathrm{Hom}}\nolimits _ B(K \otimes _ A B, R\Gamma _ Z(L) \otimes _ A B)
is a quasi-isomorphism. This follows from Lemma 47.9.7.
\square
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