counter example - affine space

Question

In the affine-n-space $\mathbb A^n_k$ (where $k$ is algebraic closed) you can define for an algebraic set $X$:

$I(X)=\left \{ f\in k[x_1,x_2,...,x_n] | \forall a \in X \,\,\,f(a)=0))\right \}$

I need to show that there is a field $k$ and sets $X$,$Y$ in which:

$I(X\cap Y)\neq I(X)+I(Y)$

(I proved earlier that $I(X\cap Y)= \sqrt{I(X)+I(Y)}$)

Answer 1

Consider for example the affine varieties $Y_1,Y_2 ⊂ \mathbb{A}^1_{\mathbb{C}}$ with ideals $I(Y_1) = (x_2 − x_1^2)$ and $I(Y_2) = (x_2)$. Their intersection $Y_1 ∩ Y_2$ is obviously the origin with ideal $I(Y_1 ∩Y_2) = (x_1,x_2)$. But $I(Y_1)+I(Y_2) = (x_2 −x_1^2,x_2) = (x_1^2,x_2)$ is not a radical ideal; only its radical is equal to $I(Y_1 ∩ Y_2) = (x_1,x_2)$. In particular we have here $$ I(Y_1 ∩Y_2)\neq I(Y_1)+I(Y_2). $$