Resolving dualities and applications to homological invariants

TL;DR

This paper characterizes dualities in resolving subcategories of modules over Artin algebras using Wakamatsu tilting bimodules, extending classical dualities and applying them to derived categories and algebraic invariants.

Contribution

It introduces a new framework for dualities via Wakamatsu tilting bimodules and extends classical dualities to Gorenstein-projective modules, with applications to derived categories and algebraic invariants.

Findings

Characterization of dualities via Wakamatsu tilting bimodules.

Extension of Miyashita's duality and Huisgen-Zimmermann's correspondence.

Invariance of higher algebraic K-groups and Ringel-Hall algebras under tilting.

Abstract

Dualities of resolving subcategories of finitely generated modules over Artin algebras are characterized as dualities with respect to Wakamatsu tilting bimodules. By restriction of these dualities to resolving subcategories of finitely generated modules with finite projective or Gorenstein-projective dimensions, Miyashita's duality and Huisgen-Zimmermann's correspondence on tilting modules as well as their Gorenstein version are obtained. Applications include constructing triangle equivalences of derived categories of finitely generated Gorenstein-projective modules and showing the invariance of higher algebraic $K$-groups and semi-derived Ringel-Hall algebras of finitely generated Gorenstein-projective modules under tilting.