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)?
Normalization of a node
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algebraic-geometry
arithmetic-geometry
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2Can you see geometrically what the normalization is? Consider first the non-completed situation, $k[x,y]/(xy)$, make a picture... – 2011-02-14
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0@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 – 2011-02-15
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0I do not understand your comment. – 2011-02-15
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0@Mariano Suarez-Alvarez geometrically it is easy: 2 disjoint lines. But why can I write it as intersection of 2 local rings? – 2011-02-17
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0@induktio: your edit substantially changed the meaning of the question. Why did you do that? – 2015-01-21
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0@AsalBeagDubh My apologies. I meant to correct the bad phrasing and the like, not to drastically alter the substance of the question. I have changed it back mathematically to what it was before, I believe, but the grammatical changes remain. – 2015-01-21
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0@AsalBeagDubh I did in my newest edit--it has to be approved first. It will be changed though. No worries. – 2015-01-21