Volume 22, issue 2 (2018)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 29
Issue 5, 2251–2782
Issue 4, 1693–2250
Issue 3, 1115–1691
Issue 2, 549–1114
Issue 1, 1–548

Volume 28, 9 issues

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
 
Subscriptions
 
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
 
Author index
To appear
 
Other MSP journals
Affine representability results in $\mathbb{A}^1$–homotopy theory, II: Principal bundles and homogeneous spaces

Aravind Asok, Marc Hoyois and Matthias Wendt

Geometry & Topology 22 (2018) 1181–1225
Bibliography
1 S Anantharaman, Schémas en groupes, espaces homogènes et espaces algébriques sur une base de dimension 1, from: "Sur les groupes algébriques", Mém. Soc. Math. France 33, Soc. Math. France (1973) 5 MR0335524
2 M Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974) 165 MR0399094
3 M Artin, A Grothendieck, J L Verdier, Théorie des topos et cohomologie étale des schémas, Tome 1 : Théorie des topos, Exposés I–IV (SGA 41 ), 269, Springer (1972) MR0354652
4 A Asok, B Doran, J Fasel, Smooth models of motivic spheres and the clutching construction, Int. Math. Res. Not. 2017 (2017) 1890 MR3658186
5 A Asok, J Fasel, Algebraic vector bundles on spheres, J. Topol. 7 (2014) 894 MR3252968
6 A Asok, J Fasel, A cohomological classification of vector bundles on smooth affine threefolds, Duke Math. J. 163 (2014) 2561 MR3273577
7 A Asok, J Fasel, Splitting vector bundles outside the stable range and 𝔸1–homotopy sheaves of punctured affine spaces, J. Amer. Math. Soc. 28 (2015) 1031 MR3369908
8 A Asok, J Fasel, Euler class groups and motivic stable cohomotopy, preprint (2016) arXiv:1601.05723
9 A Asok, M Hoyois, M Wendt, Affine representability results in 𝔸1–homotopy theory, I : vector bundles, Duke Math. J. 166 (2017) 1923 MR3679884
10 A Asok, M Hoyois, M Wendt, Generically split octonion algebras and 𝔸1–homotopy theory, preprint (2017) arXiv:1704.03657
11 A Bak, Nonabelian K–theory : the nilpotent class of K1 and general stability, K–Theory 4 (1991) 363 MR1115826
12 A Bak, R Hazrat, N Vavilov, Localization-completion strikes again : relative K1 is nilpotent by abelian, J. Pure Appl. Algebra 213 (2009) 1075 MR2498798
13 C Balwe, A Sawant, R–equivalence and 𝔸1–connectedness in anisotropic groups, Int. Math. Res. Not. 2015 (2015) 11816 MR3456703
14 C Balwe, A Sawant, 𝔸1–connectedness in reductive algebraic groups, Trans. Amer. Math. Soc. 369 (2017) 5999 MR3646787
15 J Barge, J Lannes, Suites de Sturm, indice de Maslov et périodicité de Bott, 267, Birkhäuser (2008) MR2431916
16 H Bass, E H Connell, D L Wright, Locally polynomial algebras are symmetric algebras, Invent. Math. 38 (1976) 279 MR0432626
17 P Belmans, A J de Jong, others, The Stacks project, http://stacks.math.columbia.edu, electronic reference (2005–)
18 A Białynicki-Birula, Rationally trivial homogeneous principal fibrations of schemes, Invent. Math. 11 (1970) 259 MR0276250
19 A Borel, Linear algebraic groups, 126, Springer (1991) MR1102012
20 C Cazanave, Algebraic homotopy classes of rational functions, Ann. Sci. Éc. Norm. Supér. 45 (2012) 511 MR3059240
21 J L Colliot-Thélène, M Ojanguren, Espaces principaux homogènes localement triviaux, Inst. Hautes Études Sci. Publ. Math. 75 (1992) 97 MR1179077
22 B Conrad, Reductive group schemes, from: "Autour des schémas en groupes, I", Panor. Synthèses 42/43, Soc. Math. France (2014) 93 MR3362641
23 B Conrad, O Gabber, G Prasad, Pseudo-reductive groups, 17, Cambridge Univ. Press (2010) MR2723571
24 M Demazure, A Grothendieck, Schémas en groupes, Tome II : Groupes de type multiplicatif, et structure des schémas en groupes généraux, Exposés VIII–XVIII (SGA 3II ), 152, Springer (1970) MR0274459
25 M Demazure, A Grothendieck, Schémas en groupes, Tome III : Structure des schémas en groupes réductifs, Exposés XIX–XXVI (SGA 3III ), 153, Springer (1970) MR0274460
26 D Dugger, S Hollander, D C Isaksen, Hypercovers and simplicial presheaves, Math. Proc. Cambridge Philos. Soc. 136 (2004) 9 MR2034012
27 J Fasel, Some remarks on orbit sets of unimodular rows, Comment. Math. Helv. 86 (2011) 13 MR2745274
28 J Fasel, On the number of generators of ideals in polynomial rings, Ann. of Math. 184 (2016) 315 MR3505181
29 R Fedorov, I Panin, A proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing infinite fields, Publ. Math. Inst. Hautes Études Sci. 122 (2015) 169 MR3415067
30 P Gille, Le problème de Kneser–Tits, from: "Séminaire Bourbaki, 2007/2008", Astérisque 326, Soc. Mat. de France (2009) 39 MR2605318
31 P Gille, Sur la classification des schémas en groupes semi-simples, from: "Autour des schémas en groupes, III", Panor. Synthèses 47, Soc. Math. France (2015) 39 MR3525601
32 P G Goerss, J F Jardine, Simplicial homotopy theory, 174, Birkhäuser (1999) MR1711612
33 A Grothendieck, Eléments de géométrie algébrique, IV : Étude locale des schémas et des morphismes de schémas, IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967) 5 MR0238860
34 S Hollander, A homotopy theory for stacks, Israel J. Math. 163 (2008) 93 MR2391126
35 A Iarrobino, An algebraic fibre bundle over P1 that is not a vector bundle, Topology 12 (1973) 229 MR0340254
36 J F Jardine, On the homotopy groups of algebraic groups, J. Algebra 81 (1983) 180 MR696133
37 M A Knus, Quadratic and Hermitian forms over rings, 294, Springer (1991) MR1096299
38 M I Krusemeyer, Fundamental groups, algebraic K–theory, and a problem of Abhyankar, Invent. Math. 19 (1973) 15 MR0335522
39 T Y Lam, Serre’s problem on projective modules, Springer (2006) MR2235330
40 B Margaux, The structure of the group G(k[t]) : variations on a theme of Soulé, Algebra Number Theory 3 (2009) 393 MR2525556
41 J S Milne, Étale cohomology, 33, Princeton Univ. Press (1980) MR559531
42 J Milnor, D Husemoller, Symmetric bilinear forms, 73, Springer (1973) MR0506372
43 F Morel, On the Friedlander–Milnor conjecture for groups of small rank, from: "Current developments in mathematics, 2010" (editors D Jerison, B Mazur, T Mrowka, W Schmid, R Stanley, S T Yau), International (2011) 45 MR2906371
44 F Morel, 𝔸1–algebraic topology over a field, 2052, Springer (2012) MR2934577
45 F Morel, V Voevodsky, A1–homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999) 45 MR1813224
46 L F Moser, 𝔸1–locality results for linear algebraic groups, in preparation
47 L F Moser, Rational triviale Torseure und die Serre–Grothendiecksche Vermutung, Diplomarbeit, Ludwig-Maximilians-Universität München (2008)
48 Y A Nisnevich, Espaces homogènes principaux rationnellement triviaux et arithmétique des schémas en groupes réductifs sur les anneaux de Dedekind, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984) 5 MR756297
49 I Panin, Proof of Grothendieck–Serre conjecture on principal G–bundles over regular local rings containing a finite field, preprint (2014) arXiv:1406.0247
50 I Panin, C Walter, On the motivic commutative ring spectrum BO, preprint (2010) arXiv:1011.0650
51 I Panin, C Walter, Quaternionic Grassmannians and Pontryagin classes in algebraic geometry, preprint (2010) arXiv:1011.0649
52 V A Petrov, A K Stavrova, Elementary subgroups in isotropic reductive groups, Algebra i Analiz 20 (2008) 160 MR2473747
53 D Popescu, Polynomial rings and their projective modules, Nagoya Math. J. 113 (1989) 121 MR986438
54 D Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976) 167 MR0427303
55 M S Raghunathan, Principal bundles on affine space, from: "C P Ramanujam : a tribute" (editor K G Ramanathan), Tata Inst. Fund. Res. Studies in Math. 8, Springer (1978) 187 MR541022
56 A Ramanathan, Deformations of principal bundles on the projective line, Invent. Math. 71 (1983) 165 MR688263
57 M Schlichting, Euler class groups and the homology of elementary and special linear groups, Adv. Math. 320 (2017) 1 MR3709100
58 T A Springer, F D Veldkamp, Octonions, Jordan algebras and exceptional groups, Springer (2000) MR1763974
59 A Stavrova, Homotopy invariance of non-stable K1–functors, J. K–Theory 13 (2014) 199 MR3189425
60 A A Suslin, Locally polynomial rings and symmetric algebras, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977) 503 MR0444647
61 R W Thomason, Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. in Math. 65 (1987) 16 MR893468
62 K Voelkel, M Wendt, On 𝔸1–fundamental groups of isotropic reductive groups, C. R. Math. Acad. Sci. Paris 354 (2016) 453 MR3488111
63 C A Weibel, Homotopy algebraic K–theory, from: "Algebraic K–theory and algebraic number theory" (editors M R Stein, R K Dennis), Contemp. Math. 83, Amer. Math. Soc. (1989) 461 MR991991
64 M Wendt, 𝔸1–homotopy of Chevalley groups, J. K–Theory 5 (2010) 245 MR2640204
65 M Wendt, Rationally trivial torsors in 𝔸1–homotopy theory, J. K–Theory 7 (2011) 541 MR2811715
66 G W Whitehead, On mappings into group-like spaces, Comment. Math. Helv. 28 (1954) 320 MR0065927