# In what sense are complete local rings finitely generated modules?

In the first paragraph of section 18.4 of Eisenbud's Commutative Algebra, there is the following comment.

Most interesting Noetherian rings can be written as finitely generated modules over regular subrings. For example, Noether normalization allows us to write any affine algebra as a finitely generated module over a polynomial ring of the same dimension, and every complete local ring may be written as a finitely generated module over a regular local ring of the same dimension.

What result is he alluding to in the last sentence about complete local rings? I can only think of the Cohen structure theorem, which says that every Noetherian local ring that contains a field is isomorphic to a quotient of a power series ring over a field. And certainly this is a module over $k[[x_i]]$ (a regular local ring) with a single generator. But then why would he say "finitely generated module" instead of just quotient ring? (The more general structure theorem also has a conclusion about quotient rings.) But I suppose this can't be the right result, because we don't know the dimensions match.

A Google search turns up part (c) of the theorem on page 8 of these notes. Is the result contained in any textbook or standard reference?

Edit: The answer to this question gives the location of this result in Bourbaki. A treatment in English can be found in Cohen's original article on the structure of complete local rings in Transactions of the AMS as Theorem 16. The footnote on that page notes the connection to Noether Normalization.

Also, references are given in the last few paragraphs of section 18.4.

• Here is a reference: Bourbaki, Commutative Algebra, Ch. 9, § 2, n° 5, theorem 3. – Bernard Jan 4 '16 at 0:29
• @Bernard Hi! Thank you. If you post that as an answer, I would be happy to accept it. – user4571 Jan 4 '16 at 0:32

A reference here is in Bourbaki, Commutative Algebra, ch. 9: Complete noetherian local rings, § 2 (Cohen rings), n°5, theorem 3, for the case of residual characteristic $p$.