Lemma 37.62.1 (069E)—The Stacks project

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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

$i$ is a Koszul-regular immersion in a neighbourhood of $u$ , and
$i'$ 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. $\square$

Comments (2)

Comment #8552 by Hayama Kazuma on July 08, 2023 at 04:35

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

Comment #9136 by Stacks project on May 13, 2024 at 09:42

Thanks and fixed here.

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