Question related to a stalk of a scheme

Question

Suppose $X$ is a scheme and suppose $C$ and $C'$ are two irreducible components of $X$. Suppose also that $p \in C \cap C'$. Does is it then follow that $O_{X,p}$ is not an integral domain? Thanks!

Answer 1

Yes, assuming of course $C, C'$ are distinct (which is probably what you were intending). Since the question is local, we can assume $X$ is affine (this assumption needs to be examined), say given by $Spec\ A$ for a ring $A$. If $I$ is the ideal of $C$ and $J$ is the ideal of $C'$, then $I, J$ are minimal primes of $A$ contained in the prime ideal $m$ associated to $p$. Note that $(0) \subseteq I \cap J \subsetneq I$ (since $I, J$ are distinct), and so since $I$ is a minimal prime, it follows $(0)$ is not a prime ideal, so $A$ is not a domain. It follows that $(0)A_m$ is not a prime ideal of $A_m$ (if it were, then its intersection with $A$ would also be prime), and hence $A_m=\mathcal{O}_{X,p}$ is not a domain.

If $X$ is not affine, choose an open affine $U=Spec A$ of $X$ containing $p$. The subspace $U \cap C$ is irreducible, but as pointed out in the comments, it is not clear that it is a component of $U$, and this is crucial for the proof above.