Local-global principle for congruence subgroups of Chevalley groups

• Autorzy: Himanee Apte , Alexei Stepanov
• Języki publikacji: EN

Abstrakty

EN

Suslin’s local-global principle asserts that if a matrix over a polynomial ring vanishes modulo the independent variable and is locally elementary then it is elementary. In this article we prove Suslin’s local-global principle for principal congruence subgroups of Chevalley groups. This result is a common generalization of the result of Abe for the absolute case and Apte, Chattopadhyay and Rao for classical groups. For the absolute case the localglobal principle was recently obtained by Petrov and Stavrova in the more general settings of isotropic reductive groups.

Słowa kluczowe

EN

Chevalley groups; Principal congruence subgroup; Local-global principle; Dilation principle

Kategorie tematyczne

• 20G35: Linear algebraic groups over ad\`eles and other rings and schemes

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2014
• Tom: 12
• Numer: 6
• Strony: 801-812

Daty

wydano

2014-06-01

online

2014-03-19

Twórcy

autor

Himanee Apte

• St. Petersburg State University

autor

Alexei Stepanov

Bibliografia

• [1] Abe E., Whitehead groups of Chevalley groups over polynomial rings, Comm. Algebra, 1983, 11(12), 1271–1307 http://dx.doi.org/10.1080/00927878308822906
• [2] Apte H., Chattopadhyay P., Rao R.A., A local global theorem for extended ideals, J. Ramanujan Math. Soc., 2012, 27(1), 1–20 http://dx.doi.org/10.1007/s11139-011-9299-9
• [3] Bak A., Nonabelian K-theory: The nilpotent class of K 1 and general stability, K-Theory, 1991, 4(4), 363–397 http://dx.doi.org/10.1007/BF00533991
• [4] Bak A., Hazrat R., Vavilov N., Localization-completion strikes again: relative K 1 is nilpotent by abelian, J. Pure Appl. Algebra, 2009, 213(6), 1075–1085 http://dx.doi.org/10.1016/j.jpaa.2008.11.014
• [5] Basu R., Chattopadhyay P., Rao R.A., Some remarks on symplectic injective stability, Proc. Amer. Math. Soc., 2011, 139(7), 2317–2325 http://dx.doi.org/10.1090/S0002-9939-2010-10654-8
• [6] Basu R., Rao R., Injective stability for K 1 of classical modules, J. Algebra, 2010, 323(4), 867–877 http://dx.doi.org/10.1016/j.jalgebra.2009.12.012
• [7] Chattopadhyay P., Rao R.A., Elementary symplectic orbits and improved K 1-stability, J. K-Theory, 2011, 7(2), 389–403 http://dx.doi.org/10.1017/is010002021jkt109
• [8] Grunewald F., Mennicke J., Vaserstein L., On symplectic groups over polynomial rings, Math. Z., 1991, 206(1), 35–56 http://dx.doi.org/10.1007/BF02571323
• [9] Hazrat R., Stepanov A., Vavilov N., Zhang Z., The yoga of commutators, J. Math. Sci. (N.Y.), 2011, 179(6), 662–678 http://dx.doi.org/10.1007/s10958-011-0617-y
• [10] Hazrat R., Vavilov N., Zhang Z., Relative unitary commutator calculus, and applications, J. Algebra, 2011, 343, 107–137 http://dx.doi.org/10.1016/j.jalgebra.2011.07.003
• [11] Hazrat R., Vavilov N., Zhang Z., Relative commutator calculus in Chevalley groups, J. Algebra, 2013, 385, 262–293 http://dx.doi.org/10.1016/j.jalgebra.2013.03.011
• [12] Hazrat R., Vavilov N., Zhang Z., Multiple commutator formulas for unitary groups, preprint available at http://arxiv.org/abs/1205.6866v1
• [13] Hazrat R., Zhang Z., Multiple commutator formulas, Israel J. Math., 2013, 195(1), 481–505 http://dx.doi.org/10.1007/s11856-012-0135-8
• [14] Jose S., Rao R.A., A local global principle for the elementary unimodular vector group, In: Commutative Algebra and Algebraic Geometry, Contemp. Math., 390, American Mathematical Society, Providence, 2005, 119–125 http://dx.doi.org/10.1090/conm/390/07298
• [15] Mason A.W., On subgroups of GL(n, A) which are generated by commutators. II, J. Reine Angew. Math., 1981, 322, 118–135
• [16] Petrov V.A., Stavrova A.K., Elementary subgroups in isotropic reductive groups, St. Petersburg Math. J., 2009, 20(4), 625–644 http://dx.doi.org/10.1090/S1061-0022-09-01064-4
• [17] Quillen D., Projective modules over polynomial rings, Invent. Math., 1976, 36, 167–171 http://dx.doi.org/10.1007/BF01390008
• [18] Rao R.A., Basu R., Jose S., Injective stability for K 1 of the orthogonal group, J. Algebra, 2010, 323(2), 393–396 http://dx.doi.org/10.1016/j.jalgebra.2009.09.022
• [19] Stavrova A., Homotopy invariance of non-stable K 1-functors, J. K-Theory (in press), DOI: 10.1017/is013006012jkt232
• [20] Stepanov A., Vavilov N., On the length of commutators in Chevalley groups, Israel J. Math., 2011, 185, 253–276 http://dx.doi.org/10.1007/s11856-011-0109-2
• [21] Suslin A.A., On the structure of the special linear group over polynomial rings, Math. USSR-Izv., 1977, 11(2), 221–238 http://dx.doi.org/10.1070/IM1977v011n02ABEH001709
• [22] Suslin A.A., Kopeiko V.I., Quadratic modules and the orthogonal group over polynomial rings, J. Soviet Math., 1982, 20(6), 2665–2691 http://dx.doi.org/10.1007/BF01681481
• [23] Taddei G., Normalité des groupes élémentaires dans les groupes de Chevalley sur un anneau, In: Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, Contemp. Math., 55, American Mathematical Society, Providence, 1986, 693–710 http://dx.doi.org/10.1090/conm/055.2/1862660
• [24] Tits J., Systèmes générateurs de groupes de congruence, C. R. Acad. Sci. Paris, 1976, 283(9), A693–A695
• [25] Vaserstein L.N., On the normal subgroups of GLn over a ring, In: Algebraic K-Theory, Lecture Notes in Math., 854, Springer, Berlin-New York, 1981, 456–465
• [26] Vaserstein L.N., On normal subgroups of Chevalley groups over commutative rings, Tohoku Math. J., 1986, 38(2), 219–230 http://dx.doi.org/10.2748/tmj/1178228489
• [27] Vavilov N.A., Stepanov A.V., Standard commutator formulae, Vestnik St. Petersburg Univ. Math., 2008, 41(1), 5–8 http://dx.doi.org/10.3103/S1063454108010020
• [28] Vavilov N.A., Stepanov A.V., Standard commutator formulae, revisited, Vestnik St. Petersburg Univ. Math., 2010, 43(1), 2010, 12–17 http://dx.doi.org/10.3103/S1063454110010036
• [29] Wendt M., A1-homotopy of Chevalley groups, J. K-Theory, 2010, 5(2), 245–287 http://dx.doi.org/10.1017/is010001014jkt096

Identyfikatory

DOI

10.2478/s11533-013-0391-9