Lemma 37.59.5. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

$f$ is flat and perfect, and

$f$ is flat and locally of finite presentation.

Lemma 37.59.5. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

$f$ is flat and perfect, and

$f$ is flat and locally of finite presentation.

**Proof.**
The implication (2) $\Rightarrow $ (1) is More on Algebra, Lemma 15.82.4. The converse follows from the fact that a pseudo-coherent morphism is locally of finite presentation, see Lemma 37.58.5.
$\square$

Your email address will not be published.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.

## Comments (0)