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?



Upper semicontinuity of multiplicities for finite morphisms between varieties
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.


