Polynomial extension preserves Cohen--Macaulayness

If $M=0$, then $M[X]=0$ and [F1] gives the claim vacuously. Otherwise fix $P$ in the support of $M[X]$, put $\mathfrak{p}=P\cap R$, $A=R_{\mathfrak{p}}$, $\mathfrak{m}=\mathfrak{p}A$, $B=A[X]$, $P^{\mathrm{\prime }}=PB$, and $S=B_{P^{\mathrm{\prime }}}$. The nonzero module $M_{\mathfrak{p}}$ is Cohen--Macaulay. Under the stated AC, [F4] and [F5] supply a regular parameter tuple $f_1,\dots ,f_d$ with $d=\dim M_{\mathfrak{p}}$ and nonzero finite-length quotient $N$. Polynomial extension is faithfully flat, and localization preserves injectivity. The final quotient $Q=N[X{]}_{P^{\mathrm{\prime }}}$ is nonzero: [F6] gives $N\mathrm{/}\mathfrak{m}N\mathrm{\ne }0$, and its polynomial extension localized at $P^{\mathrm{\prime }}\supseteq \mathfrak{m}B$ remains nonzero. Thus [F2] makes the tuple regular on $M[X{]}_P$.

Since $N$ is nonzero of finite length, its annihilator has radical $\mathfrak{m}$, and ${\mathrm{Supp}}_{S}Q=V(\mathfrak{m}S)$. This also follows by tensoring a finite composition series of $N$: its factors become copies of $\kappa (\mathfrak{p})[X{]}_{\bar{P}}$, where $\bar{P}=P^{\mathrm{\prime }}\mathrm{/}\mathfrak{m}B$. If $\bar{P}=0$, this is a field, so $Q$ has finite length over $S$. If $\bar{P}\mathrm{\ne }0$, write $\bar{P}=(\bar{g})$ for a monic irreducible polynomial and lift its coefficients to a monic $g\in A[X]$. Since $P^{\mathrm{\prime }}$ is the inverse image of $\bar{P}$, this lift lies in $P^{\mathrm{\prime }}$. Monicity makes multiplication by $g$ injective on $N[X]$ by highest-coefficient comparison, hence on $Q$. Moreover $Q\mathrm{/}gQ$ is supported only at the maximal ideal of $S$ (reduce modulo $\mathfrak{m}$), and is nonzero by [F6] because $Q$ is finite and nonzero and $g\in P^{\mathrm{\prime }}S$, the maximal ideal; it therefore has finite length.

In the first case the regular tuple has length $d$ and finite-length quotient; in the second, adjoining $g$ gives a regular tuple of length $d+1$ and nonzero finite-length quotient. Write its length as $l$. The definition of depth gives ${\mathrm{depth}}_{S}M[X{]}_P\ge l$, [F4] gives ${\dim }_{S}M[X{]}_P\le l$, and [F3] gives the reverse comparison between depth and dimension. Hence both equal $l$. Every localization in the support is Cohen--Macaulay, proving [F1].