Notes on De Jong's period=index theorem for central simple algebras over fields of transcendence degree two

TL;DR

This paper discusses de Jong's simplified proof of the period=index theorem for central simple algebras over fields with transcendence degree two, providing supportive lecture notes without introducing new results.

Contribution

It offers a detailed explanation of de Jong's proof and clarifies the simplified approach, serving as educational material rather than novel research.

Findings

Clarifies de Jong's proof of the period=index theorem

Provides supportive lecture notes for educational purposes

Includes a proof of the Artin splitting theorem

Abstract

These are notes on de Jong's proof of the period=index theorem over fields of transcendence degree two. They are actually about the simplified proof sketched by de Jong in the last section of his paper. These notes were meant as support for my lectures at the summer school "Central Simple Algebras over Function Fields" at the Universitat Konstanz between August, 26 and September, 1 2007 (other lectures on this subject were given by Philippe Gille, Andrew Kresch, Max Lieblich, Tamas Szamuely and Jan Van Geel). No originality is intended (except perhaps a little in the proof of the Artin splitting theorem). Various sources on which the material is based are indicated in the notes. The reader should be warned that these notes have not been updated to reflect developments in the subject which occurred after the end of the summerschool.