auslander buchsbaum serre regularity criterion

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

finite residue field projective dimension forces depth equals dimension: If the residue field of a nonzero Noetherian local ring $R$ has finite projective dimension, then $R$ is regular and $\mathrm{depth}R=\dim R=\mathrm{edim}R$.

regular local residue field projective dimension dimension: For a regular local ring $(R,\mathfrak{m},k)$ of dimension $d$, ${\mathrm{pd}}_{R}k=d$ and ${\beta }_i^R(k)=\left(\genfrac{}{}{0px}{}{d}{i}\right)$ for $0\le i\le d$, with ${\beta }_i^R(k)=0$ for $i>d$.

local global dimension equals residue field projective dimension: For a nonzero Noetherian local ring $(R,\mathfrak{m},k)$, $\mathrm{gldim}R={\mathrm{pd}}_{R}k$, allowing infinity. If this common value is $n<\mathrm{\infty }$, every $R$-module has projective dimension at most $n$.

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.

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 $0\le c\le d$.

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 $M=0$.