Exercise of commutative algebra, rational functions.

Question

This exercise is of my weekly newsletter of the subject of commutative algebra. My knowledge is restricted to the book of William Fulton, Algebraic Curves. I need help to solve it, any hints.

Exercise:

Let $Y \subseteq X$ two algebraic varieties and let $\mathcal{K}(X,Y)$ be a subset of $\mathcal{K}(X)$ that consists of rational functions defined at some point in $Y$. Prove that $\mathcal{K}(X,Y)$ is a local subring of $\mathcal{K}(X)$ and that its residual field is isomorphic to $\mathcal{K}(Y)$.

Answer 1

You want to show that some ring is a local ring. The first thing you will have to do is to find a maximal ideal.

The ring in this case is $$\mathcal K(X,Y) = \{ f \in \mathcal K(X) \mid f \text{ is defined on } Y \}.$$

What is a good candidate for a maximal ideal? HINT In a local ring, every element outside the maximal ideal is invertible. So you are looking for non-invertible elements. (mouseover the grey area for a stronger hint)

HINT What about those functions that are zero on $Y$?