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

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 valuationis 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.