In this paper we define an Euler-Poincaré characteristic which is the basis for generalizing to tame coverings of schemes the theory of the Galois module structure of rings of algebraic integers. First we define tame $G$-coverings of schemes $f: X \to Y$, where $G$ is a finite group. Then, under the assumption that the schemes are proper and of finite type over a noetherian ring $A$ and given $T$ a coherent $G$-sheaf on $X$, we define the Euler-Poincaré characteristic $\chi R\Gamma ^+(f_*(T))$, which is an element of the Grothendieck group $CT(AG)$ of all finitely generated $AG$-modules which are cohomologically trivial as $G$-modules. In fact the definition applies to certain complexes of sheaves on $X$ which occur in applications. In an appendix we include a proof of a variant of the well known Lemma of Abhyankar characterizing tame $G$-coverings of schemes.