Generalized Commutator Formulas

Communications in Algebra (Impact Factor: 0.39). 04/2011; 39(4):1441-1454. DOI: 10.1080/00927871003738964


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 [E(n)(A, I), GL(n)(A, J)] = [E(n)(A, I), E(n)(A, J)], which is a generalization of the standard commutator formular. This answers a problem posed by Stepanov and Vavilov.

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    • "This was done by L. Vaserstein in [37]. Relative commutator formula was obtained by different methods in [39] [40] [17] [15]. Relative multi-commutator formula for G = GL n was recently proved by R. Hazrat and Zhang Zuhong in [18]. "
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    • "The reason is that the first of these choices is too small as the neighbourhood on the right hand side, while the second of these choices is too large as the neighbourhood on the left hand side. The solution proposed for GL(n, R) in [33] and later applied to unitary groups in [32] consists in selecting another base of neightborhoods E(Φ, s k a) ≤ E(Φ, s k R, s k a) ≤ E(Φ, R, s k a), which is much better balanced with respect to conjugation. The following definition embodies the gist of this method. "
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    • "In fact, localisation methods used in the proof of these results have many further applications, both actual and potential: relative commutator formulas, multiple commutator formulas, nilpotency of K 1 , description of subnormal subgroups, description of various classes of overgroups, connection with excision kernels, etc. We refer to our surveys [36] [31] [32] and to our papers [29] [35] [7] [40] [37] [38] [41] [76] [39] [33] [34] for these and further applications and many further related references. "
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