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. $\square$


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