A Guide to Cohen’s Structure Theorem for Complete Local Rings

The purpose of this note is to provide for my algebra class a guide to the proof of Cohen's structure theorem for complete local rings as given by Cohen himself, in the original 1946 paper [C]. In spite of more modern and concise treatments by the likes of Nagata and Grothendieck, the original proof is a model of clarity and the entire paper is a commutative algebra masterpiece. First, a few words on the original paper. The paper is divided into three parts. Part I deals with the general theory of completions of local rings and what Cohen calls generalized local rings. The former being Noetherian commutative rings with a unique maximal ideal, the latter being quasi-local rings (R, m) with m finitely generated and satisfying Krull's intersection theorem, ∩ n≥1 m n = 0. Generalized local rings are needed in Part II when (in modern jargon) certain faithfully flat extensions of local rings are constructed in order to pass to the case where the residue field of the new ring is perfect. Using the associated graded ring, Cohen shows in Part I that the completion of a generalized local ring is a local ring (see [C; Theorem 3]). Incidently, Cohen asks whether or not generalized local rings are always local. The answer is no, but I believe that the generalized local rings he considers are local rings. Part II of the paper presents the structure theorem for complete local rings. It is interesting to note that some of the easier arguments that serve as 'base cases' for his proof either appeared in or are based on earlier work in the 1930's on valuations rings by Hasse and Schmidt, Maclane, and Teichmuller. In Part III, Cohen proves fundamental results on the stucture and ideal theory of regular local rings. Included in this section are the facts that a complete regular local ring containing a field is a power series ring over a field (a conjecture of Krull's) and any associated prime of an ideal of the principal class (i.e., height equals number of generators) in a regular local ring has the same height ('rank') as 1