# Euler Characteristic of Coverings via Sheaf Theory

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.

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Doesn't the Leray spectral sequence give this straight away? –  Gunnar Þór Magnússon Jul 12 '13 at 14:44

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).