Lemma 35.23.32. The properties

1. $\mathcal{P}(f) =$“$f$ is a Koszul-regular immersion”,

2. $\mathcal{P}(f) =$“$f$ is an $H_1$-regular immersion”, and

3. $\mathcal{P}(f) =$“$f$ is a quasi-regular immersion”

are fpqc local on the base.

Proof. We will use the criterion of Lemma 35.22.4 to prove this. By Divisors, Definition 31.21.1 being a Koszul-regular (resp. $H_1$-regular, quasi-regular) immersion is Zariski local on the base. By Divisors, Lemma 31.21.4 being a Koszul-regular (resp. $H_1$-regular, quasi-regular) immersion is preserved under flat base change. The final hypothesis (3) of Lemma 35.22.4 translates into the following algebra statement: Let $A \to B$ be a faithfully flat ring map. Let $I \subset A$ be an ideal. If $IB$ is locally on $\mathop{\mathrm{Spec}}(B)$ generated by a Koszul-regular (resp. $H_1$-regular, quasi-regular) sequence in $B$, then $I \subset A$ is locally on $\mathop{\mathrm{Spec}}(A)$ generated by a Koszul-regular (resp. $H_1$-regular, quasi-regular) sequence in $A$. This is More on Algebra, Lemma 15.32.4. $\square$

There are also:

• 2 comment(s) on Section 35.23: Properties of morphisms local in the fpqc topology on the target

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).