Derived tensor product

Definition 6.6.25 (Derived tensor product). We define the derived tensor product \[ - \otimes ^{\bL } - \colon \D ^-(\Aa ) \times \D ^-(\Aa ) \to \D ^-(\Aa ) \] as the total left derived functor of the composite \[ \Ch ^-(\Aa )\times \Ch ^-(\Aa )\xrightarrow {-\otimes -}\Ch ^-(\Aa )\longrightarrow \D ^-(\Aa ), \] where the source is equipped with the product of the projective cofibration structures of Proposition 6.6.15, Lemma 6.6.7. Indeed, on pairs of bounded below projective complexes, tensor product preserves quasi-isomorphisms, since these are chain homotopy equivalences and tensor product preserves chain homotopies in each variable. Consequently, if \(P\to C\) and \(Q\to D\) are any projective resolutions, then \[ C\otimes ^{\bL }D\simeq P\otimes Q. \] As in Lemma 6.6.19, this functor is exact in both variables, and restricts to \(\D (\Aa )_{\geq 0} \times \D (\Aa )_{\geq 0} \to \D (\Aa )_{\geq 0}\). If \(\Aa \) satisfies \((AB3)\) and \((AB4)\), then it also preserves coproducts of uniformly bounded below families in both variables.