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Let $f\colon X\to Y$ be a finite morphism between two smooth irreducible varieties over an algebraically closed field $k$. Let $x\in X$ be a (closed) point, and $y = f(x)$. Define the multiplicity of $f$ at $x$ to be $m_f(x) := \dim_k\mathcal{O}_{X,x}/M_y\mathcal{O}_{X,x}$ where here $\mathcal{O}_{X,x}$ is the local ring of $X$ at $x$, and $M_y$ is the maximal ideal in the local ring $\mathcal{O}_{Y,y}$ of $Y$ at $y$, viewed as a subring of $\mathcal{O}_{X,x}$ via the pullback $f^*$.

I have been told that the function $m_f\colon X\to\mathbb{N}$ is Zariski upper semicontinuous, but am having trouble coming up with a proof or finding a reference. Does anyone have a reference or an argument for why this is true? It seems like the natural thing to do would be to find a coherent sheaf on $X$ whose fiber at $x$ has dimension $m_f(x)$, but which sheaf?

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  • $\begingroup$ Something that looks like the arrangement of symbols: $\mathcal{O}_{X}/M\mathcal{O}_{X}$? IANAAG, though, so I have no idea if something like this is coherent. $\endgroup$ – David Roberts Sep 22 '11 at 4:34
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M. Lejeune-Jalabert and B. Teissier. Normal cones and sheaves of relative jets. Compositio Math., 28:305–331, 1974

Actually the result is much more general, not just for finite morphisms.

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