Lemma 15.65.4. Let $R \to R'$ be a flat ring map. Let $M$, $N$ be $R$-modules.

If $M$ is a finitely presented $R$-module, then $\mathop{\mathrm{Hom}}\nolimits _ R(M, N) \otimes _ R R' = \mathop{\mathrm{Hom}}\nolimits _{R'}(M \otimes _ R R', N \otimes _ R R')$.

If $M$ is $(-m)$-pseudo-coherent, then $\mathop{\mathrm{Ext}}\nolimits ^ i_ R(M, N) \otimes _ R R' = \mathop{\mathrm{Ext}}\nolimits ^ i_{R'}(M \otimes _ R R', N \otimes _ R R')$ for $i < m$.

In particular if $R$ is Noetherian and $M$ is a finite module this holds for all $i$.

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