On Characteristic Modules of Graded Quasi-hereditary Algebras

Abstract

Let G be a group and A a G-graded quasi-hereditary algebra. Then its characteristic module is proved to be G-gradable, i.e., it is isomorphic to a G-graded module as A-modules. This implies that the Ringel dual A′ of A admits a canonical G-grading which extends to the graded situation the typical equivalence between Δ-good and ∇-good modules of A and A′, respectively. It follows some consequences: the derived category of finitely generated G-graded A-modules is equivalent to the derived category of finitely generated G-graded A'-modules; if G is finite, then the Ringel dual of the smash product A#G* is isomorphic to the smash product A'#G* of A' with G.

Key Words:

• Graded quasi-hereditary algebra
• Characteristic module
• Ringel dual
• Smash product

Mathematics Subject Classification:

• 16G10
• 16G20
• 16W30

Acknowledgments

The original proof of Theorem 2.7. was done by using smash products. The present one in the paper is due to the referee which is simpler and clearer than the original one. The author is grateful to the referee for this and for his or her many other useful comments and suggestions.

The final version of the paper was completed when the author was visiting the Free University of Brussels, VUB. He would like to thank Professor Stef Caenepeel for his warm hospitality and his many helps.

The author was supported in part by Doctoral Program Foundation of Institute of Higher Education, China (2003).

Notes

^#Communicated by M. Cohen.