Cohen structure theorem

Remarks

The hard part is the existence of a coefficient field, or of a coefficient ring in the general case. Once it exists, choose generators $x_1, \dots, x_n$ of $\mathfrak m$ and send $X_i$ to $x_i$. This map from the power series ring is surjective because $R$ is complete and $\mathfrak m$ is finitely generated. In clause 2 the number of generators can be taken equal to $d$. Then the kernel is $0$, since a proper quotient of the domain $k[\![X_1, \dots, X_d]\!]$ has dimension less than $d$. A complete local ring $R$ contains a field if and only if $R$ and $k$ have the same characteristic. When $k$ is perfect of characteristic $p > 0$, the ring $C$ can be taken to be the ring of Witt vectors $W(k)$; for $k = \F_p$ this is $\Z_p$. Example: the completion of $k[x, y]_{(x, y)}/(y^2 - x^3)$ at its maximal ideal is $k[\![X, Y]\!]/(Y^2 - X^3)$.

Going deeper

For a student

Together with the faithful flatness of completion, the theorem lets one reduce questions about Noetherian local rings to quotients of power series rings, where explicit computation is possible. It implies, for example, that complete Noetherian local rings are excellent and that they are quotients of regular local rings.