A construction of dualizing categories by tensor products of categories

TL;DR

This paper demonstrates that certain tensor product constructions involving dualizing categories yield new dualizing categories with desirable properties, including the existence of almost split sequences in related categories.

Contribution

It constructs dualizing categories via tensor products of residue categories of path categories and dualizing categories, extending the class of known dualizing categories.

Findings

The tensor product of the residue category of a path category and a dualizing category is dualizing.

The category of finitely presented functors over such tensor products is dualizing and has almost split sequences.

Categories of complexes also possess almost split sequences as a consequence.

Abstract

It is shown that the idempotent completion of the additive hull of the tensor product of the residue category of the category of paths of a locally finite quiver modulo an admissible ideal and a dualizing category is dualizing. Furthermore, the category of finitely presented functors over such tensor product category is dualizing and has almost split sequences. As applications, the categories of all kinds of complexes are proved to have almost split sequences.