Lemma 31.20.1 (063C)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.20: Regular ideal sheaves
Lemma 31.20.1 (cite)

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Lemma 31.20.1. Let $X$ be a ringed space. Let $f_1, \ldots , f_ r \in \Gamma (X, \mathcal{O}_ X)$ . We have the following implications $f_1, \ldots , f_ r$ is a regular sequence $\Rightarrow$ $f_1, \ldots , f_ r$ is a Koszul-regular sequence $\Rightarrow$ $f_1, \ldots , f_ r$ is an $H_1$ -regular sequence $\Rightarrow$ $f_1, \ldots , f_ r$ is a quasi-regular sequence.

Proof. Since we may check exactness at stalks, a sequence $f_1, \ldots , f_ r$ is a regular sequence if and only if the maps

$f_ i : \mathcal{O}_{X, x}/(f_1, \ldots , f_{i - 1}) \longrightarrow \mathcal{O}_{X, x}/(f_1, \ldots , f_{i - 1})$

are injective for all $x \in X$ . In other words, the image of the sequence $f_1, \ldots , f_ r$ in the ring $\mathcal{O}_{X, x}$ is a regular sequence for all $x \in X$ . The other types of regularity can be checked stalkwise as well (details omitted). Hence the implications follow from More on Algebra, Lemmas 15.30.2, 15.30.3, and 15.30.6. $\square$

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