auslander buchsbaum base case free module

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

Projective dimension of an object: Assume projective resolutions are supplied or exist in the relevant class. The projective dimension of $M$ is $\mathrm{pd}(M)=\inf \{d\ge 0:M\text{ has a projective resolution of length }d\},$ with value $\mathrm{\infty }$ if this set is empty. A length-zero projective resolution exists exactly when $M$ is projective.

A finite flat module over a local ring is free: The standard theorem holds over arbitrary local rings; the proof written here is the Noetherian local case. Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $M$ be a finite flat $R$-module. Then $M$ is free.

Projective left and right modules are flat over an arbitrary ring: Every projective left or right module over an arbitrary ring is flat on its appropriate side.

Depth as the first nonzero Ext degree: Let $R$ be Noetherian, let $M$ be finite, and let $I$ lie in the Jacobson radical. Then ${\mathrm{depth}}_I(M)=\inf \{i\ge 0:{\mathrm{Ext}}_R^i(R\mathrm{/}I,M)\mathrm{\ne }0\},$ where the infimum of the empty set is $\mathrm{\infty }$.