Higher K-theory of Local Fields

Abstract

In this chapter we shall examine invariants of the Galois module structure on the higher algebraic K-groups of local fields. These will be constructed by the method of Example 2.1.8(i) as the Euler characteristic of suitable 2-extensions, called the local fundamental classes lying in ${\rm E}xt_{{\rm Z}\left[ {G\left( {L/K} \right)} \right]}^2\left( {{K_{2r}}\left( L \right),{K_{2r + 1}}\left( L \right)} \right)$ for r ≥ 1, where L/K is a Galois extension with group G(L/K). Actually, when L is a p-adic local field K_2r+1 (L) is not a finitely generated Z[G(L/K)]-module and so one applies the construction of Example 2.1.8(i) to the canonical corresponding element in ${\rm E}xt_{\left[ {G\left( {L/K} \right)} \right]}^2\left( {{K_{2r}}\left( L \right),{K_{2r + 1}}\left( L \right)/A} \right)$ where A is a cohomologically trivial Z[G(L/K)]-submodule chosen so that K_2r+1 (L)/A is finitely generated.