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 $N$.