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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