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Let $X,Y \subset \mathbb{A}^n$ be two irreducible affine varieties with nonempty intersection. Prove that dim$X \cap Y \geq$dim$X+$dim$Y-n$

There is a hint to use the diagonal $\Delta=\{(x,x)|x \in \mathbb{A}^n\} \subset \mathbb{A}^n \times \mathbb{A}^n$, but I don't see how this works.

Any help is appreciated!

Chi Cheuk Tsang
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    $X\cap Y$ is equal to $X\times Y\cap\Delta$. The advantage is, $\Delta$ is a complete intersection and thus you can use Krull's principal ideal theorem. You can look up this argument in Serre's Local Algebra. – Mohan May 19 '16 at 13:31
  • Thank you for your answer, I know how this is done now! – Chi Cheuk Tsang May 19 '16 at 13:47
  • @Mec see https://math.stackexchange.com/questions/3418995/reference-request-for-the-dimension-of-intersection-of-affine-varieties for a canonical answer. (This question is a duplicate.) – KReiser Jun 22 '22 at 03:25

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