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This question was given in an exercise in a course of Algebraic Geometry. I have been following the textbook of Daniel Perrin.

Question: We consider an algebraic subset $Y_1 =V(X^2 -Y) $. Show that $A/I(Y_1) \approx k[T]$. $A= k[X, Y]$

Attempt: It has been taught in one of the lectures that if $Y\subseteq \mathbb{A}^n$ is an affine algebraic set, we define the affine co-ordinate ring $A(Y)$ of $Y$ to be $A/I(Y)$.

I am aware of the definitions of the maps $V$ and $I$ but I am not able to define a map from A to $k[T]$ whose kernel is $I(Y_1)$. ( I can think of only this method to prove what is asked).

If there is any other method which is better suited, that will also be fine.

Kindly help me with this problem.

1 Answers1

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Can you draw what the set $Y_1$ looks like if $\mathsf k=\mathbb R$ say?

It may also be helpful to think about the graph of a function: if $f\colon X \to Y$ is a function, then $\Gamma(f) = \{(x,f(x)): x \in X\}$ is a subset of $X\times Y$. It is always the case that $X$ and $\text{Graph}(f)$ are isomorphic.

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