## 81.12 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 71.6.

Lemma 81.12.1. In Situation 81.2.1 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 71.6.2), Decent Spaces, Lemma 67.20.2, 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 étale 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 70.6.2. Then $u$ is a generic point of $V(f)$ by Decent Spaces, Lemma 67.20.1. Hence $\mathcal{O}_{U, u}$ has dimension $1$ by Krull's Hauptidealsatz (Algebra, Lemma 10.60.11). Thus $\xi$ is a codimension $1$ point on $X$ and $Z$ is a prime divisor as desired. $\square$

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