Divisor class group

Definition

Let $A$ be a Krull domain with field of fractions $K$, and let $X$ be the set of prime ideals of $A$ of height one. For $P \in X$, the localization $A_P$ is a discrete valuation ring; let $v_P \colon K^\times \to \Z$ be its valuation.

1. The divisor group $\operatorname{Div}(A)$ is the free abelian group on the set $X$.
2. For $x \in K^\times$, the principal divisor of $x$ is $\operatorname{div}(x) = \sum_{P \in X} v_P(x) P$; the sum is finite because $A$ is a Krull domain.
3. The map $x \mapsto \operatorname{div}(x)$ is a group homomorphism $K^\times \to \operatorname{Div}(A)$; its image is the subgroup $\operatorname{Prin}(A)$ of principal divisors.
4. The divisor class group of $A$ is the quotient group $\operatorname{Cl}(A) = \operatorname{Div}(A)/\operatorname{Prin}(A)$.