Equivariant Tamagawa numbers and Galois module theory I

Let L/K be a finite Galois extension of number fields. We use complexes ari sing from the etale cohomology of Z on open subschemes of Spec O-L to defin e a canonical element of the relative algebraic K-group K-0(Z[Gal(L/K)]. R) . We establish some basic properties of this element. and then use it to re interpret and refine conjectures of Stark, of Chinburg and of Gruenberg, Ri tter and Weiss. Our results precisely explain the connection between these conjectures and the seminal work of Bloch and Kato concerning Tamagawa numb ers. This provides significant new insight into these important conjectures and also allows one to use powerful techniques from arithmetic algebraic g eometry to obtain new evidence in their favour.