The Stacks project

Lemma 36.22.3. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes. The following are equivalent

  1. $X$ and $Y$ are tor independent over $S$, and

  2. for every affine opens $U \subset X$, $V \subset Y$, $W \subset S$ with $f(U) \subset W$ and $g(V) \subset W$ the rings $\mathcal{O}_ X(U)$ and $\mathcal{O}_ Y(V)$ are tor independent over $\mathcal{O}_ S(W)$.

  3. there exists an affine open overing $S = \bigcup W_ i$ and for each $i$ affine open coverings $f^{-1}(W_ i) = \bigcup U_{ij}$ and $g^{-1}(W_ i) = \bigcup V_{ik}$ such that the rings $\mathcal{O}_ X(U_{ij})$ and $\mathcal{O}_ Y(V_{ik})$ are tor independent over $\mathcal{O}_ S(W_ i)$ for all $i, j, k$.

Proof. Omitted. Hint: use More on Algebra, Lemma 15.61.6. $\square$

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