Kaplansky Classes and Cotorsion Theories of Complexes

Abstract

In this article we provide arguments for constructing Kaplansky classes in the category of complexes out of a Kaplansky class of modules. This leads to several complete cotorsion theories in such categories. Our method gives a unified proof for most of the known cotorsion theories in the category of complexes and can be applied to the category of quasi-coherent sheaves over a scheme as well as the category of the representations of a quiver.

Key Words:

• Artin algebra
• Derived dimension
• Representation dimension
• Triangulated category

2010 Mathematics Subject Classification:

• 16G20
• 18E30
• 18A40
• 55U35

ACKNOWLEDGMENTS

The authors would like to thank the referee for useful comments and hints that improved our exposition. We also thank Jan Šťovíček for his comments on Remark 3.6. The authors also thank the Center of Excellence for Mathematics (University of Isfahan). Part of this work has been done when the authors visited University of Bielefeld. We would like to thank Professor Henning Krause for the invitation.

The research of authors was in part supported by a grant from IPM (No. 90130216).

Notes

Communicated by I. Swanson.