minimal free resolution reduces to zero differential

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

minimal free resolution differentials land in maximal ideal: For an augmented degreewise finite free resolution over a nonzero Noetherian local ring $(R,\mathfrak{m})$, minimality means that every positive differential matrix has entries in $\mathfrak{m}$. Equivalently no positive differential admits a unit pivot, or a nonzero two-term identity direct summand. A unit pivot can be cancelled without changing the resolved module.

The balanced Tor bifunctor: For a right $R$-module $N$, a left $R$-module $M$, and $i\ge 0$, define ${\mathrm{Tor}}_i^R(N,M)$ to be either $H_i(N{\otimes }_R{P}_{\bullet })$ for a projective resolution of $M$ or $H_i(Q_{\bullet }{\otimes }_{R}M)$ for a projective resolution of $N$, identified by the preceding natural balance isomorphism. On maps it uses the homology maps induced by comparison maps; coherence makes this a well-defined covariant bifunctor.