Lemma 10.76.1. Given a flat ring map $R \to R'$ and $R$-modules $M$, $N$ the natural $R$-module map $\text{Tor}_ i^ R(M, N)\otimes _ R R' \to \text{Tor}_ i^{R'}(M \otimes _ R R', N \otimes _ R R')$ is an isomorphism for all $i$.

**Proof.**
Omitted. This is true because a free resolution $F_\bullet $ of $M$ over $R$ stays exact when tensoring with $R'$ over $R$ and hence $(F_\bullet \otimes _ R N)\otimes _ R R'$ computes the Tor groups over $R'$.
$\square$

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