Abstract
We deduce the relative version of the equivalences relating the relative Local-Global Principle and the Normality of the relative Elementary subgroups of the traditional classical groups; viz. general linear, symplectic and orthogonal groups. This generalises our previous result for the absolute case; cf. [7].
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References
H. Apte, P. Chattopadhyay, R.A. Rao, A local global theorem for extended ideals. J.Ramanujan Math. Soc. 27(1), 1–20 (2012)
H. Apte, A. Stepanov, Local-global principle for congruence subgroups of Chevalley groups. Cent. Eur. J. Math. 12(6), 801–812 (2014)
A. Bak, R. Basu, R.A. Rao, Local-Global Principle for Transvection groups, in Proceedings of the American Mathematical Society, vol. 138 (2010), pp. 1191–1204
H. Bass, Algebraic \(K\)-Theory (W. A. Benjamin Inc, New York, 1968)
R. Basu, Local-global principle for the general quadratic and the general Hermitian groups and the nilpotence of \(KH_1\), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov, (POMI) 452 (2016), Voprosy Teorii Predstavlenii Algebr i Grupp. 30, 5–31. (English version; J. Math. Sci. 232(5) (2018))
R. Basu, Results in classical algebraic \({K}\)-theory. Ph.D. thesis, Tata Institute of Fundamental Research (2007), p. 70
R. Basu, R.A. Rao, R. Khanna, On Quillen’s local global principle, in Commutative Algebra and Algebraic Geometry. Contemporary Mathematics, vol. 390 (American Mathematical Society, Providence, RI, 2005), pp. 17–30
A. Gupta, Structures over commutative rings. Ph.D. Thesis, Tata Institute of Fundamental Research (2014)
A. Gupta, A. Garge, R.A. Rao; A nice group structure on the orbit space of unimodular rows–II. J. Algebr. 407, 201–223 (2014)
R. Hazrat, N. Vavilov, \({K_1}\) of Chevalley groups are nilpotent. J. Pure Appl. Algebr. 179(1–2), 99–116 (2003)
G. Horrocks, Projective modules over an extension of a local ring. Proc. Lond. Math. Soc. 14(3), 714–718 (1964)
F. Ischebek, R.A. Rao, Ideals and Reality. Springer Monographs in Mathematics (Springer, Berlin, 2005), ISBN 3-540-23032-7
S. Jose, R. Khanna, R. Rao, S. Sharma, The quotient Unimodular Vector group is nilpotent, to appear in these Proceedings
S. Jose, R.A. Rao, A Local global principle for the elementary unimodular vector group, in Commutative Algebra and Algebraic Geometry (Bangalore, India, 2003), 119–125. Contemporary Mathematics, vol. 390 (American Mathematical Society, Providence, RI, 2005)
V.I. Kopeiko, The stabilization of Symplectic groups over a polynomial ring. Math. USSR. Sbornik 34, 655–669 (1978)
T.Y. Lam, Serre’s Conjecture. Lecture Notes in Mathematics, vol. 635 (Springer, Berlin, 1978)
J. Milnor, Introduction to Algebraic \(K\)-Theory. Annals of Mathematics Studies, vol. 72 (Princeton University Press, Princeton)
M.P. Murthy, S.K. Gupta, Suslin’s Work on Linear Groups Over Polynomial Rings and Serre’s Problem. ISI Lecture Notes, vol. 8 (Macmillan Co. of India, Ltd., New Delhi, 1980)
D. Quillen, Projective modules over polynomial rings. Invent. Math. 36, 167–171 (1976)
R.A. Rao, S.K. Yadav, Relating the principles of Quillen–Suslin theory, to appear in these Proceedings
R.A. Rao, Stably elementary homotopy. Proc. Am. Math. Soc. 137(11), 3637–3645 (2009)
J.-P. Serre, Faisceaux algébriques cohérents. (French) Ann. Math. 61(2), 197–278 (1955)
A.A. Suslin, Stably free modules. (Russian) Mat. Sb. (N.S.) 102(144)(4), 537–550 (1977)
A.A. Suslin, V.I. Kopeiko, Quadratic modules and Orthogonal groups over polynomial rings. Nauchn. Sem., LOMI 71, 216–250 (1978)
A.A. Suslin, On the structure of special linear group over polynomial rings. Math. USSR. Izv. 11, 221–238 (1977)
A.A. Suslin, The structure of the special linear group over rings of polynomials. Izv. Akad. Nauk SSSR Ser. Mat. 41, 235–252 (1977)
M.S. Tulenbaev, Schur multiplier of a group of elementary matrices of finite order. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instuta im V.A. Steklova Akad. Nauk SSSR, vol. 86 (1979), pp. 162–169
W. van der Kallen, A group structure on certain orbit sets of unimodular rows. J. Algebr. 82(2), 363–397 (1983)
L.N. Vaserstein, On the normal subgroups of \(GL_n\) over a ring, in Algebraic \(K\)-Theory, Evanston (1980) (Proc. Conf., Northwestern Univ., Evanston Ill., (1980)). Lecture Notes in Mathematics, vol. 854 (Springer, Berlin, 1981), pp. 456–465
L.N. Vaserstein, A.A. Suslin, Serre’s problem on projective modules over polynomial rings, and algebraic \(K\)-theory. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 40(5), 993–1054 (1976)
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Research by the first author was supported by SERB-MATRICS grant (File No. MTR/2017/000886) for the financial year 2018–2019.
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Basu, R., Khanna, R., Rao, R.A. (2020). The Pillars of Relative Quillen–Suslin Theory. 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_12
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DOI: https://doi.org/10.1007/978-981-15-1611-5_12
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