The Stacks project

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. $\square$

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