A <prime Weil divisor> is an integral closed subscheme of codimension one. If $Y$ has generic point $\eta_Y$, regularity in codimension one makes the local ring $\mathcal O_{X,\eta_Y}$ a <discrete valuation ring> with fraction field $K(X)$. Its normalized <discrete valuation>
$$
\nu_Y:K(X)^*\longrightarrow\mathbb Z
$$
is the order of vanishing along $Y$: writing $f=u\pi^m$ for a unit $u$ and uniformizer $\pi$ gives $\nu_Y(f)=m$. Additivity of exponents makes this a group homomorphism.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 113 / 3 / a / Solution

A prime Weil divisor is one integral codimension-one closed subscheme. At its generic point, regularity in codimension one supplies a discrete valuation and hence an order of vanishing.


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 Incoming links: Prime Weil divisor