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If $X$ is a (general) scheme and $X$ is reduced at $p$, i.e. $\mathscr{O}_{X,p}$ is reduced, does there necessarily exist an open neighborhood of $p$ on which $X$ is reduced, i.e. $\mathscr{O}_X(U)$ is reduced?

This should be true at least when $X$ is Noetherian, since one can pick generators for the nilradical of any affine neighborhood $U$ and kill them off. How generally is it true?

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This question might be of some interest and relevance: math.stackexchange.com/questions/111576/… –  Rankeya Aug 2 '12 at 0:59
    
So what comes out of the link above is that in general it is not true that if a scheme is reduced at a point, then there exists a neighborhood of that point on which it is reduced. –  Rankeya Aug 2 '12 at 1:06
    
On a positive side note, note that $X$ is reduced if and only if $\mathcal{O}_X(U)$ is reduced for all open subsets $U$ of $X$. –  Harry Aug 2 '12 at 7:29
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