Completions and Hensel’s Lemma

Abstract

In this section we shall study the completion of a ring R with respect to an ideal m, written Ȓ_m, or simply Ȓ if m is clear from the context. The construction is usually applied in the case where R is a local ring and m is the maximal ideal. If R is a polynomial ring R = k[x₁, …, x _n ] over a field, and m = (x₁, …, x _n ) is the ideal generated by the variables, then the completion is the ring k[[x₁,…, x _n ]] of formal power series over k. More generally, if k is a field and R = k[x₁, …, x _n ]/I, then the completion of R with respect to m = (x₁, …, x _n ) is the ring k[[x₁, …, x _n ]]//Ik[[x₁, …, x _n ]]. General completions can similarly be defined in terms of formal power series (Exercise 7.11), but we shall give an intrinsic development.