Equivariant Tamagawa Numbers and Galois Module Theory I

Let L/K be a finite Galois extension of number fields. We use complexes arising from the étale cohomology of Z on open subschemes of Spec O _L to define a canonical element of the relative algebraic K-group K ₀ZGal(L/K)], R. We establish some basic properties of this element, and then use it to reinterpret and refine conjectures of Stark, of Chinburg and of Gruenberg, Ritter and Weiss. Our results precisely explain the connection between these conjectures and the seminal work of Bloch and Kato concerning Tamagawa numbers. This provides significant new insight into these important conjectures and also allows one to use powerful techniques from arithmetic algebraic geometry to obtain new evidence in their favour.