# Is there a $\sum e_if_i=n$ in higher dimensions?

If $$X\to Y$$ is a finite map of connected proper algebraic curves over a field, then for any point $$y\in Y$$, the sum $$\sum e_xf_x=n$$ of ramification times inertia degrees over points $$x$$ mapping to $$y$$ equals the degree of the map.

Now if $$X\to Y$$ is any finite flat map of connected proper algebraic varieties, the Hilbert polynomial $$H(X_y)\ =\ n$$ of the fibres will be invariant since the map is flat, and constant since it's finite.

The question: Does $$H(X_y)$$ have an expression like $$\sum e_xf_x$$ as a sum of products of functions of $$x$$, each of which has a satisfying interpretation?

When $$X,Y$$ are algebraic surfaces and $$y$$ is a curve (i.e. its generic point) or a closed point, how should they be visualised?

Of course this boils down to, for a map of local rings $$(B,\mathfrak{m}_B)\to(A,\mathfrak{m}_A)$$, computing $$\dim_{A/\mathfrak{m}_A}B/\mathfrak{m}_A$$ in terms of geometric'' data. But this might not be the best way of going about it; it was suggested to me that the Grothendieck Riemann Roch formula applied to $$f_*\mathcal{O}_X$$ might give an answer. I haven't been able to find anything this way yet.

• I wasn't quite sure whether MSE or MO was more appropriate. I'd be happy for it to be migrated if I made the wrong choice. – Meow Mar 12 at 21:46