Weil divisor

Remarks

Let $Y$ be a prime divisor with generic point $\eta_Y$. The local ring $\mathcal{O}_{X, \eta_Y}$ is a discrete valuation ring, with valuation $v_Y$ on the function field $K$ of $X$. For $f \in K^\times$, the principal divisor of $f$ is $\operatorname{div}(f) = \sum_Y v_Y(f) Y$, which is a finite sum. On $\mathbb{P}^1_k$ over a field $k$, the function $f = x^2/(x - 1)$ has $\operatorname{div}(f) = 2[0] - [1] - [\infty]$. A closed point of $\mathbb{A}^2_k$ is not a prime divisor, since it has codimension two, not one. The quotient of $\operatorname{Div}(X)$ by the principal divisors is the divisor class group $\operatorname{Cl}(X)$. For example, $\operatorname{Cl}(\mathbb{P}^n_k) \cong \Z$ by the degree, for $n \ge 1$.