EQUIVARIANT EULER-POINCARE CHARACTERISTICS AND TAMENESS

In this paper we define an Euler-Poincare 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 --> 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-Poincare characteristic chiRGAMMA+(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.