If we have a Noetherian scheme $X$, is it true that for any point $p$ that is in two irreducible components of $X$, then the stalk of $X$ at $p$ is not an integral domain?
Stalk of a point in the intersection of two irreducible components of a Noetherian scheme
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algebraic-geometry
schemes
1 Answers
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The minimal primes of $\mathscr{O}_{X,p}$ are in canonical bijection with the irreducible components of $X$ passing through $p$. So, if there are two components passing through $p$, $\mathscr{O}_{X,p}$ has at least two minimal primes, and therefore cannot be a domain.
This is true whether or not $X$ is Noetherian.
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0It took me a moment to digest that; I'm sure I've seen 'minimal prime' used in some contexts for the height 1 primes, so I forgot about the prime decomposition of $\sqrt{0}$! – 2012-05-12