Proposition 6.7.7. Let \(R\) be a commutative ring. For animae \(X, Y \in \An \), there is a natural equivalence \[ C_*(X \times Y; R) \; \simeq \; C_*(X;R) \otimes ^{\bL }_R C_*(Y;R) \] in \(\D (R)\).
Proof. Chain complexes of animae are connective, so both sides are functors \(\An \times \An \to \D (R)_{\geq 0}\), and the derived tensor product of Definition 6.6.25 is defined on them. It preserves colimits in each variable: it is exact and preserves coproducts of uniformly bounded below families, and every colimit of connective objects is of this form. Similarly, the functor \(- \times -\colon \An \times \An \to \An \) preserves colimits in both variables, and \(C_*(-;R)\) preserves colimits. It follows that both sides define functors \(\An \times \An \to \D (R)_{\geq 0}\) preserving colimits in each variable. Both send \((\pt , \pt )\) to \(R[0]\), hence they are naturally equivalent by the universal property of \(\An \). □
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