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

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

2. 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.

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

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 $\mathcal{F}(U_ i)$ 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 $V$.

Proof. Let $R \to A$ be a ring map. Let $M$ be an $A$-module. If $M$ is $R$-flat, then for all primes $\mathfrak q$ the module $M_{\mathfrak q}$ is flat over $R_{\mathfrak p}$ with $\mathfrak p$ the prime of $R$ lying under $\mathfrak q$. Conversely, if $M_{\mathfrak q}$ is flat over $R_{\mathfrak p}$ for all primes $\mathfrak q$ of $A$, then $M$ is flat over $R$. See Algebra, Lemma 10.39.18. This equivalence easily implies the statements of the lemma. $\square$

## Comments (0)

There are also:

• 4 comment(s) on Section 29.25: Flat morphisms

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01U4. Beware of the difference between the letter 'O' and the digit '0'.