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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?

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up vote 5 down vote accepted

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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It 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 $\sqrt{0}$! – Hurkyl May 12 '12 at 8:12

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