Euler Characteristic of Coverings via Sheaf Theory

Question

Let $X$ be a nice space (compact $CW$-complex or triangulated space, compact manifold, whatever works), $f:Y\to X$ be a finite covering of degree $n$, and $\chi(X)$ be the euler characteristic. By the Lefschetz fixed point theorem (or by counting cells of $X$ and $Y$ with the induced cell structure) one obtains

$$\chi(Y)=n \cdot\chi(X)$$

I wonder if there is a sheaf theoretic proof of this statement, or even better a sheaf theoretic generalization of this statement by replacing the constant sheaf by any other local system.

Answer 1

Generalization: Euler characteristic with coefficients in a rank $n$ local system equals $n$ times ordinary Euler characteristic. Then using Leray (as in Gunnar Magnusson's comment) gives the case of a finite covering (and more generally that of a fibre bundle).

The following `sheafy' proof of the generalization should work. Pick an acyclic cover that trivializes the local system and use the associated Cech resolution. An essentially equivalent way would be to do an induction argument using the Mayer-Vietoris distinguished triangle.