# Construct local ring of function field variety

Hello sage community,

I want to localize a variety's field at a certain point. First I construct the field of fractions $K(V)$ for a coordinate ring $K[V]$

K.<x, y> = GF(11)[]
S = K.quotient(y^2 - x^3 - 4*x).fraction_field()


Now I wish to construct the ideal $\langle x - 2 \rangle$, but unforunately $1/(x - 2) \in S$ so $I = S$.

sage: I = S.ideal(x - 2)
sage: I
Principal ideal (1) of Fraction Field of Quotient of Multivariate Polynomial Ring in x, y over Finite Field of size 11 by the ideal (-x^3 + y^2 - 4*x)


How can I exclude all denominators from the fraction field corresponding to $x = 2, y = 4$?

That is $S$ is all $f/g \in K[V] : g(2, 4) \neq 0$.

Thank you

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I do not understand what it means to exclude denominators. Even if $g_1(2,4)\ne0$ and $g_2(2,4)\ne0$, we still may have $g_1(2,4)+g_2(2,4)=0$. That is, such polynomials do not form an ideal.

( 2022-07-30 21:07:27 +0100 )edit

Maybe I should have excluded the part about the ideal. I just want to construct the local ring, which is $K[V]_P$ aka the coordinate ring localized at a certain point. This means excluding all denominators which are a zero at P.

( 2022-07-31 07:44:58 +0100 )edit

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