Lemma 37.67.7. In Situation 37.67.1 the category of schemes flat, separated, and locally quasi-finite over the pushout $Y \amalg _ Z X$ is equivalent to the category of $(X', Y', Z', i', j', f, g, h)$ as in Lemma 37.67.6 with $f, g, h$ flat. Similarly with “flat” replaced with “étale”.

**Proof.**
If we start with $(X', Y', Z', i', j', f, g, h)$ as in Lemma 37.67.6 with $f, g, h$ flat or étale, then $Y' \amalg _{Z'} X' \to Y \amalg _ Z X$ is flat or étale by More on Algebra, Lemma 15.7.7.

For the converse, let $W \to Y \amalg _ Z X$ be a separated and locally quasi-finite morphism. Set $X' = W \times _{Y \amalg _ Z X} X$, $Y' = W \times _{Y \amalg _ Z X} Y$, and $Z' = W \times _{Y \amalg _ Z X} Z$ with obvious morphisms $i', j', f, g, h$. Form the pushout $Y' \amalg _{Z'} X'$. We obtain a morphism

of schemes over $Y \amalg _ X Z$ by the universal property of the pushout. If we do not assume that $W \to Y \amalg _ Z X$ is flat, then in general this morphism won't be an isomorphism. (In fact, More on Algebra, Lemma 15.6.5 shows the displayed arrow is a closed immersion but not an isomorphism in general.) However, if $W \to Y \times _ Z X$ is flat, then it is an isomorphism by More on Algebra, Lemma 15.7.7. $\square$

## 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Your email address will not be published.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.

## Comments (0)

There are also: