Patching and local-global principles for gerbes over semi-global fields with an application to homogeneous spaces Open Access

Abstract

Starting in 2007, Harbater and Hartmann introduced a new patching setup for semi-global fields, establishing a patching framework for vector spaces, central simple algebras, quadratic forms and other algebraic structures. In subsequent work with Krashen, the patching framework was refined and extended to torsors and certain Galois cohomology groups. After describing this framework, we will discuss an extension of the patching equivalence to bitorsors and gerbes. Building up on these results, we then proceed to derive a characterisation of a local-global principle for gerbes and bitorsors in terms of factorization. These results can be expressed in the form of a Mayer-Vietoris sequence in non-abelian hypercohomology with values in the crossed-module G → Aut(G). After proving the local-global principle for certain bitorsors and gerbes using the characterization mentioned above, we conclude with an application on rational points for homogeneous spaces via a study of the associated quotient stack.

Table of Contents

1 Introduction 1

1.1 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2

2 Grothendieck topologies, descent and stacks 5

2.1 Grothendieck Topologies . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Categories fibered in groupoids . . . . . . . . . . . . . . . . . . . . . 8

2.3 Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Algebraic groups 20

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Classification of split semisimple algebraic groups . . . . . . . . . . . 25

3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Torsors and principal homogeneous spaces . . . . . . .30

3.5  Etale Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5.1 Torsors and Cohomology . . . . . . . . . . . . . . . . . . . . . 34

3.5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Non-abelian hypercohomology 40

4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Properties of hypercohomology . . . . . . . . . . . . . . . . . . . .43

5 Patching and local global principles 46

5.1 Introduction to field patching . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Example: Patching over arithmetic curves . . . . . . . . . . . . 52

5.2.1 Patching over the projective line . . . . . . . . . . . . . . . . . 52

5.2.2 The local case . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Patching for torsors and a local-global principle . . . . . . . . . .. 68

5.3.1 Factorization and local-global in the local case . . .  . . . 72

5.4 Separable Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.5 Galois Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6 Bitorsors 78

6.1 A semi-cocyclic description . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2 Patching for bitorsors . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.3 Bitorsor Factorization. . . . . . . . . . . . . . . . . . . . . . . . . .90

7 Gerbes 96

Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.2 A semi-cocyclic description . . . . . . . . . . . . . . . . . . . . . . 103

7.3 Patching for gerbes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.4 A Mayer-Vietoris sequence and a local-global principle for gerbes . . . 112

8 Arithmetic Curves 115

8.1 Bitorsors Patching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.2 Bitorsor Factorization. . . . . . . . . . . . . . . . . . . . . .116

8.3 Gerbe Patching and Mayer-Vietoris . . . . . . . . . . . . . . . 123

8.4 Local-global principles for gerbes . . . . . . . . . . . . . . . . 125

9 Homogeneous Spaces 126

Bibliography 132

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School	Laney Graduate School
Department	Mathematics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Mathematics
Keyword	Mayer-Vietoris patching algebraic geometry local-global
Committee Chair / Thesis Advisor	Raman, Parimala, Emory University
Committee Members	Zureick-Brown, David, Emory University Venapally, Suresh, Emory University

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