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How do you describe the normalization of $\mathrm{Spec}(k[[x,y]]/(xy))$? In particular, call this $N$. Why can $N$ be written as the intersection of 2 local rings and the same holds for the maximal ideal (with 2 maximal ideals)?

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Can you see geometrically what the normalization is? Consider first the non-completed situation, $k[x,y]/(xy)$, make a picture... –  Mariano Suárez-Alvarez Feb 14 '11 at 17:37
@Mariano Suarez-Alvarez is it something like this $k[x]\oplus k[y] $??this is not normal! I have to take $Spec(k[x])\cup Spec(k[y])$ ans normalization –  unknown Feb 15 '11 at 9:26
I do not understand your comment. –  Mariano Suárez-Alvarez Feb 15 '11 at 16:45
@Mariano Suarez-Alvarez geometrically it is easy: 2 disjoint lines. But why can I write it as intersection of 2 local rings? –  unknown Feb 17 '11 at 10:16
@induktio: your edit substantially changed the meaning of the question. Why did you do that? –  user64687 Jan 21 at 15:10

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