Derived tensor product in the bounded above setting

For bounded-above right and left $R$-complexes $N$ and $M$, supply bounded-above projective replacements $P_N\to N$ and $P_M\to M$, and the homotopy lifts required for their model functors. The derived tensor product $N{\otimes }_R^{\mathbf{L}}M\in D^{-}(\mathbf{A}\mathbf{b})$ is the object represented by $\mathrm{Tot}(P_N{\otimes }_{R}M)$, equivalently $\mathrm{Tot}(N{\otimes }_R{P}_M)$. If existence of enough module projectives is invoked, assume AC as in Module categories have enough projectives.

Use the cochain reindexing of The tensor product of a right and a left chain complex is totalized by direct sums with the Koszul differential. Projectives are flat by Projective left and right modules are flat over an arbitrary ring. Consequently Bounded above flat tensor complexes preserve quasi isomorphisms gives the natural quasi-isomorphisms $P_N{\otimes }_R{P}_M\to P_N{\otimes }_{R}M$ and $P_N{\otimes }_R{P}_M\to N{\otimes }_R{P}_M$. This fixes the balancing identification. Homotopy comparison maps between projective replacements, as in Existence of the bounded above left total derived functor, induce tensor maps; chain homotopies induce total homotopies with the same Koszul rule. Quasi-isomorphisms in the other variable are inverted by the flat-tensor lemma, so localization gives a bifunctor $D^{-}(\mathrm{Mod}\text{\hspace{0.17em}⁣}\text{-}R)\times D^{-}(R\text{\hspace{0.17em}⁣}\text{-}\mathrm{Mod})\to D^{-}(\mathbf{A}\mathbf{b})$, independent of the supplied representatives up to these comparisons.