The Stacks project

Lemma 15.64.4. Let $R \to R'$ be a flat ring map. Let $M$, $N$ be $R$-modules.

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

  2. 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 $i$.

Proof. By Algebra, Lemma 10.73.1 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 Lemma 15.64.3 we conclude (1) and (2) holds. The final statement follows from this and Lemma 15.63.17. $\square$


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