K-theoretic Background

The aim of these lectures is to cover the requisite background from Algebraic K-theory needed to prove the main result of Soule [So79, Theoreme 5], which is dealt with in detail in [Li15]. This result of Soule which, along with an observation of Schneider, proves the vanishing of the etale cohomology group H-et(2)(Spec Z[1/l], Q(l)/Z(l) (m)), m >= 2, is equivalent to the finiteness of H-et(2)(Spec Z[1/l], Z(l)(m)) and plays a crucial role in the proof of the Tamagawa number conjecture due to [BK90]. In this article, I will denote an odd prime, all the rings considered will be commutative, associative, unital rings, unless otherwise mentioned, and all categories are small. We shall mainly consider Quillen's K-theory referring the reader to [Q73a] for details. We would like to thank the referee for the comments which helped improve the exposition.