residue field splits off reduced maximal ideal

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

regular element reduction preserves minimal resolution: Let $(R,\mathfrak{m},k)$ be nonzero Noetherian local, let $M$ be a nonzero finite module, and let $x\in \mathfrak{m}$ be a nonzerodivisor on both $R$ and $M$. Reducing a minimal free resolution of $M$ modulo $x$ gives a minimal free resolution of $M\mathrm{/}xM$ over $S=R\mathrm{/}(x)$. Moreover ${\mathrm{pd}}_S(M\mathrm{/}xM)={\mathrm{pd}}_{R}M$, including infinity. For $M=0$ the zero-complex assertion also holds, with both projective dimensions zero.

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 $\mathrm{pd}K=n-1$.

Projective dimension at most n iff higher Ext vanishes: Assume the Axiom of Dependent Choice. In an abelian category with enough projectives and enough injectives, fix supplied projective and injective resolution data on all objects. Let $M$ be an object and $n\ge 0$. The following are equivalent: 1. $\mathrm{pd}(M)\le n$; 2. ${\mathrm{Ext}}^k(M,N)=0$ for every object $N$ and every $k>n$; 3. ${\mathrm{Ext}}^{n+1}(M,N)=0$ for every object $N$.

The canonical iterated free cover gives projective resolution data under AC (The iterated free-module resolution is canonical in ZF). Module categories are Grothendieck (Module categories are Grothendieck categories), and under AC their functorial injective embeddings can be iterated to give injective resolutions (Grothendieck abelian categories have functorial injective embeddings, Every module admits an injective resolution).