Duality and Serre functor in homotopy categories

For a (right and left) coherent ring $A$, we show that there exists a duality between homotopy categories {\mathbb{K}}^{{\rm{b}}}({\rm mod}{\mbox{-}}A^{{\rm op}}) and {\mathbb{K}}^{{\rm{b}}}({\rm mod}{\mbox{-}}A). If $A=\Lambda$ is an artin algebra of finite global dimension, this duality restricts to a duality between their subcategories of acyclic complexes, {\mathbb{K}}^{{\rm{b}}}_{\rm ac}({\rm mod}{\mbox{-}}\Lambda^{\rm op}) and {\mathbb{K}}^{{\rm{b}}}_{\rm ac}({\rm mod}{\mbox{-}}\Lambda). As a result, it will be shown that, in this case, {\mathbb{K}}_{\rm ac}^{{\rm{b}}}({\rm mod}{\mbox{-}}\Lambda) admits a Serre functor and hence has Auslander-Reiten triangles.Comment: arXiv admin note: text overlap with arXiv:1605.0474