Cohen–Macaulayness over a finite regular local base

Given: The objects and hypotheses in the example. We work with the Axiom of Choice; cited dependent-choice and resolution-existence hypotheses are retained.

auslander buchsbaum formula: For a nonzero finite module $M$ of finite projective dimension over a nonzero Noetherian local ring $R$, ${\mathrm{pd}}_{R}M+{\mathrm{depth}}_{R}M=\mathrm{depth}R$. Consequently such an $M$ with $\mathrm{depth}M=\mathrm{depth}R$ is free.

auslander buchsbaum serre regularity criterion: For a nonzero Noetherian local ring $(R,\mathfrak{m},k)$ the following are equivalent: $R$ is regular; ${\mathrm{pd}}_{R}k<\mathrm{\infty }$; $\mathrm{gldim}R<\mathrm{\infty }$; and every finite $R$-module has finite projective dimension. When these hold, $\mathrm{gldim}R={\mathrm{pd}}_{R}k=\dim R$. A nonzero finite module over regular local $R$ is maximal Cohen–Macaulay (depth $\dim R$) if and only if it is free.

Injective integral extensions preserve Krull dimension: Assume the Axiom of Choice. Let $A\subseteq B$ be an injective integral extension of nonzero commutative rings. Then $\dim A=\dim B$.

Every system of parameters is regular in a Cohen--Macaulay module: Every system of parameters of a nonzero finite Cohen--Macaulay module over a Noetherian local ring is a regular sequence on that module.

One regular system of parameters implies Cohen--Macaulayness: Let $0\mathrm{\ne }M$ be finite over a Noetherian local ring. If one system of parameters for $M$ is $M$-regular, then $M$ is Cohen--Macaulay. Here a system of parameters for $M$ means a tuple $x_1,\dots ,x_d$ in the maximal ideal, where $d=\dim {\mathrm{Supp}}_R(M)$, such that $M\mathrm{/}(x_1,\dots ,x_d)M$ has finite length.

regular local rings are domains and cohen macaulay: A regular local ring $R$ of dimension $d$ is a domain and Cohen–Macaulay. For every regular system $(x_1,\dots ,x_d)$, the tuple is $R$-regular and $R\mathrm{/}(x_1,\dots ,x_c)$ is regular local of dimension $d-c$ for all $0\le c\le d$.

Depth is bounded by support dimension: For every nonzero finite module $M$ over a Noetherian local ring $R$, $0\le {\mathrm{depth}}_R(M)\le \dim {\mathrm{Supp}}_R(M)\mathrm{.}$ The nonzero hypothesis is essential for this formulation: under the adopted convention ${\mathrm{depth}}_R(0)=+\mathrm{\infty }$, whereas the empty support has no nonnegative Krull dimension.

Assuming the Axiom of Choice, Nakayama's lemma: Assume the Axiom of Choice. Let $R$ be a commutative ring, let $I⊴R$ satisfy $I\subseteq J(R)$, and let $M$ be a finitely generated left $R$-module. If $IM=M$, then $M=0$.