Lemma 29.25.2. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-modules. The following are equivalent

The sheaf $\mathcal{F}$ is flat over $S$.

For every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the $\mathcal{O}_ S(V)$-module $\mathcal{F}(U)$ is flat.

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 modules $\mathcal{F}|_{U_ i}$ is flat over $V_ j$, for all $j\in J, i\in I_ j$.

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 $\mathcal{F}(U_ i)$ is a flat $\mathcal{O}_ S(V_ j)$-module, for all $j\in J, i\in I_ j$.

Moreover, if $\mathcal{F}$ is flat over $S$ then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $\mathcal{F}|_ U$ is flat over $V$.

## Comments (0)

There are also: