Complexes with the derived double centralizer property

Abstract

In representation theory, the double centralizer property is an important property for a module (bimodule). In this paper, we extend this property to complexes in derived categories of algebras, under the name derived double centralizer property. Let A be a finite dimensional algebra over an algebraically closed field. Characterizations for complexes of finitely generated A-modules with the derived double centralizer property and for tilting complexes in the bounded derived category with the endomorphism algebras hereditary are given. In particular, when A is a upper triangular matrix algebra, all complexes of A-A-bimodules with this property are classified.

Keywords:

• Complexes
• derived categories
• double centralizer property
• hereditary algebras

2020 Mathematics Subject Classification:

• 16E05
• 16E35
• 16E60

Acknowledgments

This work was supervised by professor Alexander Zimmermann while the author was a visiting PhD student at the Université de Picardie. The author would like to thank him for many helpful discussions. He also wants to thank professor Bernhard Keller for some valuable comments, professor Amnon Yekutieli for the reminding of the reference [18] and the referee for useful suggestions which improved the article.

Additional information

Funding

The author was supported by the China Scholarship Council and the Natural Science Foundation of China (No. 12171213).