Pseudo-dualizing complexes and pseudo-derived categories
Leonid Positselski
TL;DR
This work develops a comprehensive interpolating framework between dualizing/dedualizing theory and derived categories by introducing pseudo-dualizing complexes and corresponding pseudo-derived categories. It constructs and analyzes lower/upper pseudo-coderived and pseudo-contraderived categories via Auslander and Bass classes, proving triangulated equivalences that extend the classical co-contra and derived Morita correspondences. Under finiteness and coherence hypotheses, it also treats dualizing/relative dualizing complexes, base change, and deconstructibility to ensure Hom-sets and adjoint functors, thereby unifying several layers of derived, coderived, and contraderived perspectives. The results illuminate base-change behavior and relative duality, with potential applications to representation theory and geometric contexts such as ind-schemes, while generalizing Rickard-type Morita theory to infinitely generated and non-Noetherian settings.
Abstract
The definition of a pseudo-dualizing complex is obtained from that of a dualizing complex by dropping the injective dimension condition, while retaining the finite generatedness and homothety isomorphism conditions. In the specific setting of a pair of associative rings, we show that the datum of a pseudo-dualizing complex induces a triangulated equivalence between a pseudo-coderived category and a pseudo-contraderived category. The latter terms mean triangulated categories standing "in between" the conventional derived category and the coderived or the contraderived category. The constructions of these triangulated categories use appropriate versions of the Auslander and Bass classes of modules. The constructions of derived functors providing the triangulated equivalence are based on a generalization of a technique developed in our previous paper arXiv:1503.05523.