Colocalization functors in derived categories and torsion theories

TL;DR

This paper generalizes Benson's method for computing colocalizations in derived categories, providing explicit formulas for hereditary torsion classes on rings, which enhances understanding of torsion theories and derived functors.

Contribution

It extends Benson's construction to arbitrary hereditary torsion classes, offering a general explicit formula for colocalization in derived categories.

Findings

Provides an explicit formula for colocalization with respect to hereditary torsion classes.

Generalizes Benson's results from group rings to arbitrary rings.

Enhances computational tools for derived categories and torsion theories.

Abstract

Let R be a ring and let T be a hereditary torsion class of R-modules. The inclusion of the localizing subcategory generated by T into the derived category of R has a right adjoint, which is a colocalization. Benson has recently shown how to compute this right sdjoint when R is the group ring of a finite group over a prime field and T is the hereditary torsion class generated by a simple module. We generalize Benson's construction to the case where T is any hereditary torsion class on R. This yields an explicit formula for the colocalization with respect to T, using an injective cogenerator for T.