Proposition 6.7.10. For an anima \(X \in \An \) and a bounded below chain complex \(A \in \D ^-(\Z )\), there is a natural equivalence \[ C_*(X;A) \,\simeq \, C_*(X;\Z ) \otimes ^{\bL }_{\Z } A \] in \(\D ^-(\Z )\).
Proof. Say \(A\) is concentrated in degrees \(\geq m\). Both \(C_*(-;A)\) and \(C_*(-;\Z ) \otimes ^{\bL }_{\Z } A\) then take values in \(\D (\Z )_{\geq m}\), which is closed under colimits in \(\D (\Z )\). Both preserve colimits: for the first this is the defining property from Construction 6.2.12, and for the second it follows as in Proposition 6.7.7, since \(C_*(-;\Z )\) is colimit-preserving with connective values and \(-\otimes ^{\bL }_{\Z }A\) is exact and preserves coproducts of uniformly bounded below families. Finally, both send \(\pt \) to \(A\). By the universal property of \(\An \), they are naturally equivalent. □
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