Local ring embeds in rational functions on incident components (${\mathcal{O}}_{X,P}\hookrightarrow \mathrm{Rat}(Y)$)

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 $Q(B)=\mathrm{Rat}(Y)$. One cannot generally replace $Y$ by a single component through an intersection point.