The balanced Tor bifunctor

Assume Dependent Choice (DC), as in The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain, and work with supplied projective resolutions of the modules under consideration. 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 left and right projective constructions of Tor are naturally isomorphic. On module maps use the well-defined homology maps from Each resolution-defined Tor construction is covariant in both variables. The natural coherent identifications of The Tor balance isomorphism is natural and coherent under a change of resolutions make this a covariant bifunctor up to canonical change-of-resolution isomorphism, not literal equality between different resolution complexes.

DC licenses the comparison maps and homotopies in those suppliers. Existence of projective resolutions for arbitrary modules is a separate prerequisite, not a consequence asserted here of DC. When such existence and a system of resolutions have separately been supplied, this gives the usual Tor bifunctor on the corresponding module categories. Without DC, homology of a specified tensor-resolution complex is still defined; claiming the balanced bifunctor then requires explicit comparison and coherence data instead of silently invoking the DC-bearing theorems.