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

KEYWORDS:

• Artin algebra
• derived category
• duality
• functor category

1991 MATHEMATICS SUBJECT CLASSIFICATION:

• 18E30
• 16E35
• 18G25

Acknowledgments

We would like to thank the referee for her/his useful comments and hints that improved our exposition. The first two authors also thank the Center of Excellence for Mathematics (University of Isfahan). Part of this work is done while the first author were visiting Max-Planck Institute for Mathematics-Bonn (MPIM). He would like to thank MPIM for support and the excellent atmosphere.