= Prime Weil divisor
A prime Weil divisor on a <Noetherian> integral <scheme> $X$ is an integral closed subscheme of <codimension of an algebraic subvariety>[codimension] one. Its <generic point> $\eta$ has a one-dimensional <local ring> $\mathcal O_{X,\eta}$.
If $X$ is a <normal variety>, this local ring is a one-dimensional <Noetherian> <normal domain>, hence a <discrete valuation ring>. More generally it suffices that this local ring be regular. A <uniformizer> $\pi$ then writes every nonzero <rational function> as $u\pi^m$ with $u$ a unit, giving its integer order along the divisor. This is the valuation used in the <principal Weil divisor> construction.<The Stacks project>[Lemma 10.119.7, tag 00PD: equivalence of DVR, one-dimensional regular local ring and one-dimensional Noetherian normal local domain]{ref}
The codimension-one definition alone does not imply regularity. On the cusp $\operatorname{Spec}k[t^2,t^3]$, the closed point $(t^2,t^3)$ is a prime Weil divisor. Its local ring has a maximal ideal requiring two generators, as its quotient by the square has basis $t^2,t^3$. It is therefore not a <discrete valuation ring>. Passing to the <normalization> $k[t]$ supplies a different local ring with uniformizer $t$; normality is a substantive hypothesis, not part of the definition of every prime divisor.
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