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Affine representability results in \(\mathbb A^1\)-homotopy theory. II: Principal bundles and homogeneous spaces. (English) Zbl 1400.14061
This paper is the second one of a series where the authors continue to work on the theme of their previous paper [Duke Math. J. 166, No. 10, 1923–1953 (2017; Zbl 1401.14118)]. Their aim is to extend the relation between vector bundles over smooth affine schemes and the general linear group (or Grassmannians) in the (unstable) \(\mathbb{A}^1\)-homotopy category established in loc. cit. to more general group schemes as well as homogeneous spaces. In Theorem 2.3.5, they extend the affine representability result in loc. cit. to general group schemes using a similar method. In Theorem 2.4.2, they prove the affine representability for homogeneous spaces. In Theorem 3.3.7, they prove an analogue of the Bass-Quillen conjecture for isotropic reductive algebraic groups over infinite fields (see Definition 3.3.5). They find several applications of these results such as oriented bundles, symplectic bundles, affine quadrics \(Q_{2n-1}\) and nonstable \(K\)-theory.

MSC:
14F42 Motivic cohomology; motivic homotopy theory
55R15 Classification of fiber spaces or bundles in algebraic topology
14L10 Group varieties
20G15 Linear algebraic groups over arbitrary fields
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References:
[1] ; Anantharaman, Sur les groupes algébriques. Mém. Soc. Math. France, 33, 5, (1973)
[2] 10.1007/BF01390174 · Zbl 0317.14001
[3] ; Artin, Théorie des topos et cohomologie étale des schémas, Tome 1 : Théorie des topos, Exposés I-IV (SGA 41 ). Lecture Notes in Math., 269, (1972) · Zbl 0234.00007
[4] 10.1093/imrn/rnw065 · Zbl 1405.14050
[5] 10.1112/jtopol/jtt046 · Zbl 1326.14098
[6] 10.1215/00127094-2819299 · Zbl 1314.14044
[7] 10.1090/S0894-0347-2014-00818-3 · Zbl 1329.14045
[8] 10.1215/00127094-0000014X · Zbl 1401.14118
[9] 10.1007/BF00533991 · Zbl 0741.19001
[10] 10.1016/j.jpaa.2008.11.014 · Zbl 1167.19002
[11] 10.1093/imrn/rnv044 · Zbl 1349.14150
[12] 10.1090/tran/7090 · Zbl 1364.14017
[13] 10.1007/978-3-7643-8710-5
[14] 10.1007/BF01403135 · Zbl 0371.13007
[15] 10.1007/BF01404652 · Zbl 0213.47301
[16] 10.1007/978-1-4612-0941-6
[17] 10.24033/asens.2172 · Zbl 1419.14025
[18] 10.1007/BF02699492 · Zbl 0795.14029
[19] ; Conrad, Autour des schémas en groupes, I. Panor. Synthèses, 42/43, 93, (2014)
[20] 10.1017/CBO9780511661143
[21] ; Demazure, Schémas en groupes, Tome II : Groupes de type multiplicatif, et structure des schémas en groupes généraux, Exposés VIII-XVIII (SGA 3II ). Lecture Notes in Math., 152, (1970) · Zbl 0209.24201
[22] ; Demazure, Schémas en groupes, Tome III : Structure des schémas en groupes réductifs, Exposés XIX-XXVI (SGA 3III ). Lecture Notes in Math., 153, (1970) · Zbl 0212.52810
[23] 10.1017/S0305004103007175 · Zbl 1045.55007
[24] 10.4171/CMH/216 · Zbl 1205.13013
[25] 10.4007/annals.2016.184.1.3 · Zbl 1372.13014
[26] 10.1007/s10240-015-0075-z · Zbl 1330.14077
[27] ; Gille, Séminaire Bourbaki, 2007/2008. Astérisque, 326, 39, (2009)
[28] ; Gille, Autour des schémas en groupes, III. Panor. Synthèses, 47, 39, (2015)
[29] 10.1007/978-3-0348-8707-6
[30] 10.1007/BF02732123
[31] 10.1007/s11856-008-0006-5 · Zbl 1143.14003
[32] 10.1016/0040-9383(73)90008-6 · Zbl 0268.14005
[33] 10.1016/0021-8693(83)90215-6 · Zbl 0521.20026
[34] 10.1007/978-3-642-75401-2
[35] 10.1007/BF01418849 · Zbl 0247.14005
[36] 10.1007/978-3-540-34575-6
[37] 10.2140/ant.2009.3.393 · Zbl 1178.20043
[38] ; Milne, Étale cohomology. Princeton Mathematical Series, 33, (1980)
[39] 10.1007/978-3-642-88330-9
[40] ; Morel, Current developments in mathematics, 2010, 45, (2011)
[41] 10.1007/978-3-642-29514-0 · Zbl 1263.14003
[42] 10.1007/BF02698831 · Zbl 0983.14007
[43] ; Nisnevich, C. R. Acad. Sci. Paris Sér. I Math., 299, 5, (1984)
[44] ; Petrov, Algebra i Analiz, 20, 160, (2008)
[45] 10.1017/S0027763000001288 · Zbl 0663.13006
[46] 10.1007/BF01390008 · Zbl 0337.13011
[47] ; Raghunathan, C P Ramanujam : a tribute. Tata Inst. Fund. Res. Studies in Math., 8, 187, (1978)
[48] 10.1007/BF01393340 · Zbl 0492.14007
[49] 10.1016/j.aim.2017.08.034 · Zbl 1387.19002
[50] 10.1007/978-3-662-12622-6
[51] 10.1017/is013006012jkt232 · Zbl 1314.19002
[52] ; Suslin, Izv. Akad. Nauk SSSR Ser. Mat., 41, 503, (1977)
[53] 10.1016/0001-8708(87)90016-8 · Zbl 0624.14025
[54] 10.1016/j.crma.2016.01.026 · Zbl 1387.14065
[55] 10.1090/conm/083/991991
[56] 10.1017/is010001014jkt096 · Zbl 1200.14039
[57] 10.1017/is011004020jkt157 · Zbl 1228.19002
[58] 10.1007/BF02566938 · Zbl 0058.16801
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