Lemma 37.62.1. Let $S$ be a scheme. Let $U$, $P$, $P'$ be schemes over $S$. Let $u \in U$. Let $i : U \to P$, $i' : U \to P'$ be immersions over $S$. Assume $P$ and $P'$ smooth over $S$. Then the following are equivalent

1. $i$ is a Koszul-regular immersion in a neighbourhood of $u$, and

2. $i'$ is a Koszul-regular immersion in a neighbourhood of $u$.

Proof. Assume $i$ is a Koszul-regular immersion in a neighbourhood of $u$. Consider the morphism $j = (i, i') : U \to P \times _ S P' = P''$. Since $P'' = P \times _ S P' \to P$ is smooth, it follows from Divisors, Lemma 31.22.9 that $j$ is a Koszul-regular immersion, whereupon it follows from Divisors, Lemma 31.22.12 that $i'$ is a Koszul-regular immersion. $\square$

Comment #8552 by Hayama Kazuma on

typo: “in a neighbourhood of $x$”, here, $x$ is $u\in U$?

There are also:

• 4 comment(s) on Section 37.62: Local complete intersection morphisms

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 069E. Beware of the difference between the letter 'O' and the digit '0'.