Lemma 36.22.4. Let $X \to S$ and $Y \to S$ be morphisms of schemes. Let $S' \to S$ be a morphism of schemes and denote $X' = X \times _ S S'$ and $Y' = Y \times _ S S'$. If $X$ and $Y$ are tor independent over $S$ and $S' \to S$ is flat, then $X'$ and $Y'$ are tor independent over $S'$.

Proof. Omitted. Hint: use Lemma 36.22.3 and on affine opens use More on Algebra, Lemma 15.61.4. $\square$

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