Generalized Commutator Formulas

Abstract

Let A be an algebra which is a direct limit of module finite algebras over a commutative ring R with 1. Let I, J be two-sided ideals of A, GL _n (A, I) the principal congruence subgroup of level I in GL _n (A), and E _n (A, I) the relative elementary subgroup of level I. Using Bak's localization-patching method, we prove the commutator formula

which is a generalization of the standard commutator formular. This answers a problem posed by Stepanov and Vavilov.

Key Words:

• Congruence subgroups
• Elementary subgroup
• General linear group
• Localization and patching
• Normal structure
• Sandwich classification theorems

2000 Mathematics Subject Classification:

• 20G35
• 20G15

ACKNOWLEDGMENTS

The first author acknowledges the support of EPSRC first grant scheme EP/D03695X/1. The second author acknowledges the support of NSFC (Grant 10971011). The authors thank Nikolai Vavilov for suggesting the topic of the article to them.

Notes

Communicated by V. A. Artamonov.