I'm studyng divisors from Basic Algebraic Geometry of Shafarevich. He takes an irreducible quasi-projective variety $X$, regular in codimension $1$ and claims that for any codimension-one irreducible subvariety $C\subset X$ there exists an affine open $U\subset X$ such that $C\cap U$ is defined by the local equation $\pi\in k[U]$. If I have correctly understood, this means that the prime ideal of regular functions on $U$ vanishing along $C\cap U$ is generated by $\pi$ and $C\cap U=\lbrace \pi=0\rbrace$.
Now, for any $f\in k[U]$, I can define a valuation $$\nu_C(f):=\textit{the integer $k\geq 0$ s.t. $f\in (\pi^k)\setminus(\pi^{k+1})$}.$$ I would check that the valuation $\nu_C(f)$ does not depend from $U$. The author prooceds in this way.
I don't understand why the ideal of regular functions on $V$ vanishing along $C\cap V$ is generated by $\pi_{|V}$.
Any help?
The ideal of regular functions on $V$ vanishing along $C\cap V$ is a localization of the ideal of regular functions on $U$ vanishing along $C\cap U$. If a set generates an ideal, then it also generates any localization of that ideal: an arbitrary element in a localization of an ideal is just a fraction with numerator from the original ideal and denominator a unit, and so if you can generate the numerator, you have succeeded.