Colocalizing subcategories and cosupport

TL;DR

This paper classifies certain subcategories in various triangulated categories, introduces a notion of cosupport, and advances understanding of local homology with potential for future applications.

Contribution

It provides a classification of colocalizing subcategories and develops a new framework for local homology and cosupport in triangulated categories.

Findings

Classification of Hom closed colocalizing subcategories of the stable module category.

Classification of colocalizing subcategories of the homotopy category of injectives over an exterior algebra.

Development of a notion of local homology and cosupport for triangulated categories.

Abstract

The Hom closed colocalizing subcategories of the stable module category of a finite group are classified. Along the way, the colocalizing subcategories of the homotopy category of injectives over an exterior algebra, and the derived category of a formal commutative differential graded algebra, are classified. To this end, and with an eye towards future applications, a notion of local homology and cosupport for triangulated categories is developed, building on earlier work of the authors on local cohomology and support.