The Stacks project

Definition 41.9.3. (See Morphisms, Definition 29.25.1). Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module.

  1. Let $x \in X$. We say $\mathcal{F}$ is flat over $Y$ at $x \in X$ if $\mathcal{F}_ x$ is a flat $\mathcal{O}_{Y, f(x)}$-module. This uses the map $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ to think of $\mathcal{F}_ x$ as a $\mathcal{O}_{Y, f(x)}$-module.

  2. Let $x \in X$. We say $f$ is flat at $x \in X$ if $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is flat.

  3. We say $f$ is flat if it is flat at all points of $X$.

  4. A morphism $f : X \to Y$ that is flat and surjective is sometimes said to be faithfully flat.

Comments (0)

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Your email address will not be published.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 0253. Beware of the difference between the letter 'O' and the digit '0'.