The Stacks project

80.12 Preparation for principal divisors

This section is the analogue of Chow Homology, Section 42.16. Some of the material in this section partially overlaps with the discussion in Spaces over Fields, Section 70.6.

Lemma 80.12.1. In Situation 80.2.1 let $X/B$ be good. Assume $X$ is integral.

  1. If $Z \subset X$ is an integral closed subspace, then the following are equivalent:

    1. $Z$ is a prime divisor,

    2. $|Z|$ has codimension $1$ in $|X|$, and

    3. $\dim _\delta (Z) = \dim _\delta (X) - 1$.

  2. If $Z$ is an irreducible component of an effective Cartier divisor on $X$, then $\dim _\delta (Z) = \dim _\delta (X) - 1$.

Proof. Part (1) follows from the definition of a prime divisor (Spaces over Fields, Definition 70.6.2), Decent Spaces, Lemma 66.20.2, and the definition of a dimension function (Topology, Definition 5.20.1).

Let $D \subset X$ be an effective Cartier divisor. Let $Z \subset D$ be an irreducible component and let $\xi \in |Z|$ be the generic point. Choose an ├ętale neighbourhood $(U, u) \to (X, \xi )$ where $U = \mathop{\mathrm{Spec}}(A)$ and $D \times _ X U$ is cut out by a nonzerodivisor $f \in A$, see Divisors on Spaces, Lemma 69.6.2. Then $u$ is a generic point of $V(f)$ by Decent Spaces, Lemma 66.20.1. Hence $\mathcal{O}_{U, u}$ has dimension $1$ by Krull's Hauptidealsatz (Algebra, Lemma 10.59.10). Thus $\xi $ is a codimension $1$ point on $X$ and $Z$ is a prime divisor as desired. $\square$

Comments (0)

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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0EPF. Beware of the difference between the letter 'O' and the digit '0'.