auslander buchsbaum formula

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

auslander buchsbaum base case free module: If a nonzero finite module $M$ over a nonzero Noetherian local ring $R$ has projective dimension zero, then it is finite free of positive rank and ${\mathrm{depth}}_{R}M=\mathrm{depth}R$.

auslander buchsbaum projective dimension one: If $M$ is a nonzero finite module of projective dimension one over a nonzero Noetherian local ring $R$, then $\mathrm{depth}R\ge 1$ and $\mathrm{depth}M=\mathrm{depth}R-1$.

auslander buchsbaum syzygy projective dimension: Let $0\to K\to F_0\to M\to 0$ be the initial minimal presentation of a nonzero finite module over a nonzero Noetherian local ring. If $0<n=\mathrm{pd}M<\mathrm{\infty }$, then $K\mathrm{\ne }0$ and $\mathrm{pd}K=n-1$.

auslander buchsbaum first syzygy depth: In a minimal presentation $0\to K\to F\to M\to 0$ of a nonzero finite module over a nonzero Noetherian local ring, let $n=\mathrm{pd}M\ge 2$ be finite. If $\mathrm{depth}K=\mathrm{depth}R-(n-1)$, then $\mathrm{depth}M=\mathrm{depth}K-1$.