Abstract
Jose–Rao introduced and studied the Special Unimodular Vector group \(\mathrm{SUm_r(R)}\) and \(\mathrm{EUm_r(R)}\), its Elementary Unimodular Vector subgroup. They proved that for \(r \ge 2\), \(\mathrm{EUm_r(R)}\) is a normal subgroup of \(\mathrm{SUm_r(R)}\). The Jose–Rao theorem says that the quotient Unimodular Vector group, \(\mathrm{SUm_r(R)}/\mathrm{EUm_r(R)}\), for \(r \ge 2\), is a subgroup of the orthogonal quotient group \(\mathrm{SO}_{2(r+1)}(R)/{\mathrm{EO}}_{2(r + 1)}(R)\). The latter group is known to be nilpotent by the work of Hazrat–Vavilov, following methods of A. Bak; and so is the former. In this article we give a direct proof, following ideas of A. Bak, to show that the quotient Unimodular Vector group is nilpotent of class \(\le d = \dim (R)\). We also use the Quillen–Suslin theory, inspired by A. Bak’s method, to prove that if \(R = A[X]\), with A a local ring, then the quotient Unimodular Vector group is abelian.
Keywords
2010 Mathematics Subject Classification
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Acknowledgements
The second author thanks the Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India, for the funding of project MTR/2017/000875 under Mathematical Research Impact Centric Support (MATRICS).
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Khanna, R., Jose, S., Sharma, S., Rao, R.A. (2020). The Quotient Unimodular Vector Group is Nilpotent. In: Ambily, A., Hazrat, R., Sury, B. (eds) Leavitt Path Algebras and Classical K-Theory. Indian Statistical Institute Series. Springer, Singapore. https://doi.org/10.1007/978-981-15-1611-5_13
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DOI: https://doi.org/10.1007/978-981-15-1611-5_13
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