Duality and Serre functor in homotopy categories

Abstract

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 $A=Λ$ is an artin algebra of finite global dimension, this duality restricts to a duality between their subcategories of acyclic complexes, ${\mathbb{K}}^{\rm{b}}_{\rm ac}({\rm mod}{\mbox{-}}Λ^{\rm op})$ and ${\mathbb{K}}^{\rm{b}}_{\rm ac}({\rm mod}{\mbox{-}}Λ).$ As a result, it will be shown that, in this case, ${\mathbb{K}}_{\rm ac}^{\rm{b}}({\rm mod}{\mbox{-}}Λ)$ admits a Serre functor and hence has Auslander-Reiten triangles.