Lemma 29.25.3. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

1. The morphism $f$ is flat.

2. For every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the ring map $\mathcal{O}_ S(V) \to \mathcal{O}_ X(U)$ is flat.

3. There exists an open covering $S = \bigcup _{j \in J} V_ j$ and open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that each of the morphisms $U_ i \to V_ j$, $j\in J, i\in I_ j$ is flat.

4. There exists an affine open covering $S = \bigcup _{j \in J} V_ j$ and affine open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that $\mathcal{O}_ S(V_ j) \to \mathcal{O}_ X(U_ i)$ is flat, for all $j\in J, i\in I_ j$.

Moreover, if $f$ is flat then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_ U : U \to V$ is flat.

Proof. This is a special case of Lemma 29.25.2 above. $\square$

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