= Local ring embeds in rational functions on incident components
{title2=$\mathcal O_{X,P}\hookrightarrow\operatorname{Rat}(Y)$}
Let $Y$ be the reduced union of the <irreducible components> of an <affine variety> $X$ containing $P$. Removing all other components leaves an open neighbourhood of $P$ on which $X$ and $Y$ agree, so their <local rings> at $P$ agree. In the <coordinate ring> $B=k[Y]$, every denominator not vanishing at $P$ avoids all minimal primes: each component of $Y$ contains $P$. Such a denominator is a <non-zero-divisor>. Consequently $B_{\mathfrak m_P}$ injects into the <total quotient ring> $Q(B)=\operatorname{Rat}(Y)$. One cannot generally replace $Y$ by a single component through an intersection point.
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