Remark 15.63.21. Let $R$ be ring map. Let $M$, $N$ be $R$-modules. Let $R \to R'$ be a flat ring map. By Algebra, Lemma 10.72.1 we have $\mathop{\mathrm{Ext}}\nolimits ^ i_{R'}(M \otimes _ R R', N \otimes _ R R') = \mathop{\mathrm{Ext}}\nolimits ^ i_ R(M, N \otimes _ R R')$. Combined with Remark 15.63.20 we conclude that

$\mathop{\mathrm{Hom}}\nolimits _ R(M, N) \otimes _ R R' = \mathop{\mathrm{Hom}}\nolimits _{R'}(M \otimes _ R R', N \otimes _ R R')$

if $M$ is a finitely presented $R$-module and that

$\mathop{\mathrm{Ext}}\nolimits ^ i_ R(M, N) \otimes _ R R' = \mathop{\mathrm{Ext}}\nolimits ^ i_{R'}(M \otimes _ R R', N \otimes _ R R')$

is an isomorphism for $i < m$ if $M$ is $(-m)$-pseudo-coherent. In particular if $R$ is Noetherian and $M$ is a finite module this holds for all $i$.

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