The Stacks project

82.12 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 72.6.

Lemma 82.12.1. In Situation 82.2.1 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 72.6.2), Decent Spaces, Lemma 68.20.2, 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 71.6.2. Then $u$ is a generic point of $V(f)$ by Decent Spaces, Lemma 68.20.1. 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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