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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)