INTRODUCTION

Let K/k be a finite algebraic extension and let | | be a valuation on k. We do not suppose that k is complete and ask ourselves what extensions, if any, there are of | | to K. We shall encounter this problem also when | | is archimedean. For most of this chapter we can consider arch, and non-arch, valuations together. The situation we shall be considering is usually evoked by the keyword semi-local.

Suppose that the valuation | |1 on K extends | | and let K1 be the completion of K with respect to it. Then K1 contains the completion k of k with respect to | |. A basis {Bi} of K/k clearly generates K1 as a k-vector space. There is, however, no reason to expect that the Bi, considered as elements of K1, will be linearly independent over k, and we can conclude only that [K1 : k] ≤ [K : k]. Multiplication gives K1 a natural structure as K-module.

We shall also require the tensor product k ⊗k, K. This can be described concretely, if non-canonically, as follows. Let B1,…,Bn be a basis for K/k. There are cijℓ ∈ k such that

Then k ⊗k, K is an n-dimensional k-vector space with a basis which we identify with the B1:

It has a ring structure, multiplication being defined by (1.1) and by k-linearity. We identify K in k ⊗k, K with the linear combinations of the Bi with coefficients in k.