On compactly generated torsion pairs and the classification of co-t-structures for commutative noetherian rings

TL;DR

This paper classifies co-t-structures in the derived category of commutative noetherian rings, develops a related theory for Hom-orthogonal pairs, and characterizes silting objects, revealing limited co-t-structure types.

Contribution

It introduces a theory for compactly generated Hom-orthogonal pairs in triangulated categories, extending Bousfield localization, and classifies co-t-structures for certain rings.

Findings

Only trivial and canonical co-t-structures exist for perfect complexes over connected rings.

The developed theory parallels Bousfield localization in triangulated categories.

Complete classification of silting objects in the derived category.

Abstract

We classify compactly generated co-t-structures on the derived category of a commutative noetherian ring. In order to accomplish that, we develop a theory for compactly generated Hom-orthogonal pairs (also known as torsion pairs in the literature) in triangulated categories that resembles Bousfield localization theory. Finally, we show that the category of perfect complexes over a connected commutative noetherian ring admits only the trivial co-t-structures and (de)suspensions of the canonical co-t-structure and use this to describe all silting objects in the category.