Field Patching and Galois Cohomology Pubblico

Abstract

Field Patching and Galois Cohomology
By Feng Chen

Let T be a complete discrete valuation ring with uniforrmizer t, and X a smooth projective curve over S = SpecT. Let F = K(X) be the function field and let F = K (X) be the completion of F with respect to the discrete valuation defined by the closed fibre X.

In this paper, we construct indecomposable and noncrossed product division algebras over F. This is done by defining an index-preserving homomorphism and using this map s to lift indecomposable and noncrossed product division algebras over F to indecomposable and noncrossed product division algebras over F, respectively.

Table of Contents

I Introduction...1

1 Introduction...2

II Background...5

2 Division Algebras and Brauer Groups...6

2.1 Definitions and Basic Facts...6
2.2 Index and Period...8
2.3 Noncrossed Product Division Algebras...9
2.4 Indecomposable Division Algebras...11

3 Patching Over Fields...13

3.1 Notation...13
3.2 Main Results from Patching over Fields...15

III The Main Construction...18

4 Splitting Map...19

4.1 Construction over an Open Affine Subset...20
4.2 Construction over Closed Points...22
4.3 The Map is Well Defined...25
4.4 s Splits the Restriction Map...27

5 The Splitting Map Preserves Index...31

5.1 Index Computation Over Affine Open Set...33
5.2 Index Computation Over Closed Points...35

6 Indecomposable and Noncrossed Product Division Algebras over Curves over Complete Discrete Valuation Rings...39

6.1 Indecomposable Division Algebras over F...40
6.2 Noncrossed Products over F...41

Bibliography...43

About this Dissertation

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics
Parola chiave	field patching Galois cohomology indecomposable division algebra noncrossed product division algebra
Committee Chair / Thesis Advisor	Brussel, Eric, Emory University
Committee Members	Raman, Parimala, Emory University Powers, Victoria, Emory University Venapally, Suresh , Emory University

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