Stable functors of derived equivalences and Gorenstein projective modules

TL;DR

This paper introduces stable functors derived from non-negative triangle functors between derived categories, establishing their properties and applications to Gorenstein projective modules and homological conjectures.

Contribution

It generalizes previous work by defining stable functors with good properties and shows their role in connecting derived equivalences to stable categories of Gorenstein projective modules.

Findings

Stable functors have exactness and compositional properties.

Derived equivalences induce triangle equivalences between Gorenstein projective stable categories.

Applications include shorter proofs of known homological conjectures.

Abstract

From certain triangle functors, called non-negative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are certain additive functors between the stable categories of the abelian categories. The construction generalizes a previous work by Hu and Xi. We show that the stable functors of non-negative functors have nice exactness property and are compatible with composition of functors. This allows us to compare conveniently the homological properties of objects linked by the stable functors. Particularly, we prove that the stable functor of a derived equivalence between two arbitrary rings provides an explicit triangle equivalence between the stable categories of Gorenstein projective modules. This generalizes a result of Y. Kato. Our results can also be applied to provide shorter proofs of some known results on homological conjectures.