Grade of maximal ideals in polynomial rings over Artinian local rings

Question

If $R$ is a commutative Artinian ring it is well-known that $R$ is Cohen-Macaulay. Also, if $S$ is a Cohen-Macaulay ring, then any polynomial ring $S[X_1,\dots,X_n]$ is so. Now if $R$ is a commutative Artinian local ring, how we could derive that the grade of each maximal ideal of $R[X_1,\dots,X_n]$ equals $n$?

Answer 1

The problem reduces to the following:

If $R$ is artinian, then any maximal ideal in $R[X_1,\dots,X_n]$ has height $n$.

Let $M$ be a maximal ideal in $R[X_1,\dots,X_n]$. Then $M\cap R=m$, where $m$ is a maximal ideal of $R$, hence $m[X_1,\dots,X_n]\subseteq M$. Now look at $$R[X_1,\dots,X_n]/m[X_1,\dots,X_n]\simeq (R/m)[X_1,\dots,X_n]$$ which is (isomorphic to) a polynomial ring over field. The maximal ideal $M/m[X_1,\dots,X_n]$ has height $n$ (in a polynomial ring over a field this is a well known property), hence height of $M$ is at least $n$, that is, equals $n$. (Note that $\dim R[X_1,\dots,X_n]=n$.)