Lemma 31.14.7 (01WZ)—The Stacks project

Processing math: 0%

Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.14: Effective Cartier divisors and invertible sheaves
Lemma 31.14.7 (cite)

Lemma 31.14.7. Let $X$ be a locally ringed space. Let $f \in \Gamma (X, \mathcal{O}_ X)$ . The following are equivalent:

$f$ is a regular section, and
for any $x \in X$ the image $f \in \mathcal{O}_{X, x}$ is a nonzerodivisor.

If $X$ is a scheme these are also equivalent to

for any affine open $\mathop{\mathrm{Spec}}(A) = U \subset X$ the image $f \in A$ is a nonzerodivisor,
there exists an affine open covering $X = \bigcup \mathop{\mathrm{Spec}}(A_ i)$ such that the image of $f$ in $A_ i$ is a nonzerodivisor for all $i$ .

Proof. Omitted. $\square$

Comments (0)

There are also:

2 comment(s) on Section 31.14: Effective Cartier divisors and invertible sheaves

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Your email address will not be published.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.