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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 $\mathcal{K}(Y)$.

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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 $Y$?

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