Cohen Structure Theorem

The Cohen structure theorem completely characterizes the structure of complete local rings, showing they are all quotients of power series rings. This is the deepest structural result in the theory of local rings.

Coefficient Fields and Rings

Definition

Let $(R, \mathfrak{m}, k)$ be a complete local ring. A coefficient field is a subfield $K \subseteq R$ that maps isomorphically to $k = R/\mathfrak{m}$ under the natural projection. A coefficient ring is a complete local subring $(V, pV) \subseteq R$ that maps surjectively onto $k$ with $\ker = pV$ for some prime $p$ (this is a complete discrete valuation ring if $\operatorname{char}(k) = p > 0$ and $\operatorname{char}(R) = 0$ ).

Theorem7.5Cohen Structure Theorem

Let $(R, \mathfrak{m}, k)$ be a complete Noetherian local ring.

Equal characteristic case: If $\operatorname{char}(R) = \operatorname{char}(k)$ , then $R$ contains a coefficient field $K \cong k$ , and there is a surjection $K[[x_1, \ldots, x_n]] \twoheadrightarrow R$ for some $n$ .
Mixed characteristic case: If $\operatorname{char}(R) = 0$ and $\operatorname{char}(k) = p > 0$ , then $R$ contains a coefficient ring $V$ (a complete DVR with residue field $k$ and uniformizer $p$ ), and there is a surjection $V[[x_1, \ldots, x_n]] \twoheadrightarrow R$ .

In both cases, $R \cong S/I$ where $S$ is a regular local ring (power series ring) and $I$ is an ideal.

Consequences

ExampleComplete regular local rings

By Cohen's theorem, a complete regular local ring of equal characteristic is isomorphic to $k[[x_1, \ldots, x_d]]$ where $d = \dim R$ . In mixed characteristic, a complete regular local ring of dimension $d$ is isomorphic to $V[[x_1, \ldots, x_{d-1}]]$ where $V$ is a complete DVR.

ExampleClassification of complete DVRs

A complete discrete valuation ring $V$ with residue field $k$ is either:

$k[[t]]$ (equal characteristic), or
A finite extension of $\mathbb{Z}_p$ or $W(k)$ (the Witt vectors of $k$ ) in mixed characteristic

This classification is fundamental in algebraic number theory and $p$ -adic Hodge theory.

RemarkAlgebraic geometry interpretation

Cohen's theorem says every complete local ring is the quotient of a smooth (regular) local ring. Geometrically, this means every formal neighborhood of a point on a variety can be presented as the zero locus of some equations in a formal polydisc. This is the formal analogue of the fact that every variety locally embeds in affine space.