Reflexivity in Derived Categories

TL;DR

This paper explores how adjoint functors between abelian categories extend to derived categories, characterizing reflexive complexes and identifying the largest class of objects where a Cotilting Theorem applies.

Contribution

It provides a detailed description of reflexive complexes in derived categories and links these to objects in the original abelian categories, extending Cotilting Theorem applicability.

Findings

Reflexive complexes are characterized in derived categories.

Objects reflexive as stalk complexes form the largest class for Cotilting Theorem.

Results apply to functors with any finite cohomological dimension.

Abstract

An adjoint pair of contravariant functors between abelian categories can be extended to the adjoint pair of their derived functors in the associated derived categories. We describe the reflexive complexes and interpret the achieved results in terms of objects of the initial abelian categories. In particular we prove that, for functors of any finite cohomological dimension, the objects of the initial abelian categories which are reflexive as stalk complexes form the largest class where a Cotilting Theorem in the sense of Colby and Fuller works.