R-equivalence and norm principles in algebraic groups Pubblico
Bhaskhar, Nivedita (2016)
Published
Abstract
We start by exploring the theme of R-equivalence in algebraic groups. First introduced by Manin to study cubic surfaces, this notion proves to be a fundamental tool in the study of rationality of algebraic group varieties. A k-variety is said to be rational if its function field is purely transcendental over k. We exploit Merkurjev's fundamental computations of the R-equivalence classes of adjoint classical groups and give a recursive construction to produce an infinite family of non-rational adjoint groups coming from quadratic forms living in various levels of the filtration of the Witt group. This extends the earlier results of Merkurjev and P. Gille where the forms considered live in the first and second level of the filtration. In a different direction, we address Serre's injectivity question which asks whether a principal homogeneous space under a connected linear algebraic group admitting a zero cycle of degree one in fact has a rational point. We give a positive answer to this question for any smooth connected reductive k-group whose Dynkin diagram contains connected components only of type An, Bn or Cn. We also investigate Serre's question for reductive k-groups whose derived subgroups admit quasi-split simply connected covers. We do this by relating Serre's question to the norm principles previously proved by Barquero and Merkurjev. The study of norm principles are interesting in their own right and we examine in detail the case of groups of the non-trialitarian Dn type and get a scalar obstruction defined up to spinor norms whose vanishing will imply the norm principle for these groups. This in turn will also yield a positive answer to Serre's question for all connected reductive k-groups of classical type.
Table of Contents
1 An introduction. 1
1.1 The story. 1
1.2 The plan. 5
1.3 The prerequisites. 6
1.3.1 Quadratic form theory. 6
1.3.2 Algebras with involutions. 9
2 Examining Clifford's algebras. 11
2.1 Generalizing quaternions. 11
2.2 Clifford algebra of a quadratic form. 13
2.2.1 The even Clifford algebra C0(V,q). 15
2.2.2 Structure theorem for Clifford algebras. 16
2.2.3 Involutions on Clifford algebras. 18
2.3 Clifford algebra of an algebra with orthogonal involution. 20
2.4 Examples serve better than description. 25
3 A walk through the classification of linear algebraic groups. 26
3.1 Some adjectives of linear algebraic groups. 26
3.2 Classical groups a la Weil. 28
3.3 Classification of classical groups. 30
3.3.1 Case I: Simply connected. 30
3.3.2 Case II: Adjoint. 31
3.4 Type Dn details. 32
3.4.1 The Clifford bimodule. 32
3.4.2 The Clifford group. 33
3.4.3 The vector representation. 35
3.4.4 The Spin group. 37
3.4.5 The extended Clifford group. 37
4 A crash course on group cohomology. 39
4.1 The Ext functor. 40
4.1.1 Recipes for computing cohomology. 41
4.1.2 Connectingmaps. 44
4.2 A nice ZG projective resolution of Z. 44
4.2.1 Rewriting the complex F'(Pi). 46
4.3 Cocycles and coboundaries in low dimensions. 47
4.4 Special morphisms between cohomology groups. 49
4.4.1 Restrictions and inflations. 49
4.4.2 Coinduction and corestriction. 51
4.4.3 Gratuities. 53
4.5 A computation: Hilbert 90. 55
5 A glimpse of Galois cohomology. 58
5.1 Profinite groups. 58
5.2 Profinite group cohomology. 60
5.3 Non-abelian cohomology. 61
5.3.1 A principal homogeneous space. 62
5.3.2 Two long exact sequences. 63
5.3.3 Three computations. 65
6 Reviewing R-equivalence and rationality. 69
6.1 What is being rational?. 70
6.2 Detecting non-rationality. 72
6.3 Early examples of non-rational groups. 76
6.3.1 Chevalley's example. 76
6.3.2 Serre's example. 78
6.4 The story for simply connected groups. 79
6.4.1 The reduced Whitehead of an algebra. 81
6.4.2 Positive rationality results for simply connected groups. 83
6.5 The story for adjoint groups. 84
6.5.1 Merkurjev's forumula. 85
6.5.2 Merkurjev's example. 87
6.6 A natural question. 90
7 More examples of non-rational groups. 92
7.1 Notations and Conventions. 93
7.2 Strategy. 93
7.3 Lemmata. 94
7.4 Comparison of some Hyp groups. 97
7.5 A recursive procedure. 102
7.6 Conclusion. 105
8 On norm principles. 106
8.1 What is a normprinciple?. 107
8.2 Classical examples. 108
8.3 More norm principles. 109
9 On a question of Serre. 112
9.1 Serre's question. 112
9.1.1 Known results. 114
9.2 Preliminaries. 115
9.2.1 Reduction to characteristic 0. 115
9.2.2 Lemmata. 117
9.3 Serre's question and norm principles. 119
9.3.1 Pushouts. 119
9.3.2 Intermediate groups G^ and G~. 121
9.3.3 Relating Serre's question and norm principle. 123
9.4 Quasi-split groups. 125
10 Obstruction to norm principles for groups of type Dn. 128
10.1 Preliminaries. 128
10.1.1 The map u* for n odd. 130
10.1.2 The map u for n even. 130
10.2 An obstruction to being in the image of u* for n odd. 131
10.3 An obstruction to being in the image of u for n even. 135
10.4 Scharlau's norm principle revisited. 137
10.5 Spinor obstruction to norm principle for non-trialitarian Dn. 139
11 A summary. 141
Bibliography. 144
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