Cohen structure theorem for artinian local rings

Question

Let $(R,m)$ be an artinian local ring. Since $m^n=0$ for some $n$, it is clear that $R$ is complete with respect to $m$-adic topology. Now i want to know that how do we state the Cohen structure theorem for $R$?

Answer 1

$\DeclareMathOperator{\char}{\operatorname{char}}$$\DeclareMathOperator{\m}{\mathfrak{m}}$$\DeclareMathOperator{\n}{\mathfrak{n}}$The Cohen structure theorem states that any Noetherian complete local ring $(R, \m)$ is a quotient of a power series ring over a nice ring, in some number of variables. The number of variables is related to $\mu_R(\m)$, the minimal number of generators of $\m$ (the so-called embedding dimension of $R$). To be precise, setting $n := \mu_R(\m)$:

i) If $\char R = \char R/\!\m$ (the equicharacteristic case), $R$ has a coefficient field $k \subseteq R$ (i.e. the composite $k \hookrightarrow R \twoheadrightarrow R/\!\m$ is an isomorphism), and $R \cong k[[t_1, \ldots, t_n]]/I$ for some ideal $I \subseteq k[[t_1, \ldots, t_n]]$.

ii) If $\char R \ne \char R/\!\m$ (which can only happen if $\char R/\!\m =: p > 0$), then $R$ still has a coefficient ring $(V, \n)$ which is a complete DVR with $V/\!\n \cong R/\!\m$, and $R \cong V[[t_1, \ldots, t_n]]/I$ for some $I$. Now if $p \not \in \m^2$ (the unramified case), one can decrease the number of variables by $1$: $R \cong V[[t_1, \ldots, t_{n-1}]]/I$ for some $I$.

As user26857 mentions, one has in the above that $\dim R = 0$ iff $I$ is primary to the maximal ideal of the power series ring.