Lemma 76.51.1. Let $S$ be a scheme. Consider a commutative diagram of algebraic spaces

$\xymatrix{ Z' \ar[d] \ar[r] & Y' \ar[d] \\ X' \ar[r] & B' }$

over $S$. Let $B \to B'$ be a morphism. Denote by $X$ and $Y$ the base changes of $X'$ and $Y'$ to $B$. Assume $Y' \to B'$ and $Z' \to X'$ are flat. Then $X \times _ B Y$ and $Z'$ are Tor independent over $X' \times _{B'} Y'$.

Proof. By Derived Categories of Spaces, Lemma 75.20.3 we may check tor independence étale locally on $X \times _ B Y$ and $Z'$. This1 reduces the lemma to the case of schemes which is More on Morphisms, Lemma 37.69.1. $\square$

[1] Here is the argument in more detail. Choose a surjective étale morphism $W' \to B'$ with $W'$ a scheme. Choose a surjective étale morphism $W \to B \times _{B'} W'$ with $W$ a scheme. Choose a surjective étale morphism $U' \to X' \times _{B'} W'$ with $U'$ a scheme. Choose a surjective étale morphism $V' \to Y' \times _{B'} W'$ with $V'$ a scheme. Observe that $U' \times _{W'} V' \to X' \times _{B'} Y'$ is surjective étale. Choose a surjective étale morphism $T' \to Z' \times _{X' \times _{B'} Y'} U' \times _{W'} V'$ with $T'$ a scheme. Denote $U$ and $V$ the base changes of $U'$ and $V'$ to $W$. Then the lemma says that $X \times _ B Y$ and $Z'$ are Tor independent over $X' \times _{B'} Y'$ as algebraic spaces if and only if $U \times _ W V$ and $T'$ are Tor independent over $U' \times _{W'} V'$ as schemes. Thus it suffices to prove the lemma for the square with corners $T', U', V', W'$ and base change by $W \to W'$. The flatness of $Y' \to B'$ and $Z' \to X'$ implies flatness of $V' \to W'$ and $T' \to U'$.

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