Galois Descent and Severi-Brauer Varieties

Eric Brussel
Emory University
Spring 2009



This is an expository paper, aimed at graduate students who have had some algebraic geometry. We cover the basic theory of Galois descent in categories of vector spaces with tensor (e.g., quadratic spaces, algebras, commutative algebras, and central simple algebras), then in categories of affine schemes, quasi-projective varieties, quasi-coherent sheaves, and locally free sheaves of fixed rank. We apply the theory to prove the main results about Severi-Brauer varieties, following Artin. This is by now a ``classical'' subject, at least at this level, and we prove nothing new. Our presentation is essentially a mashup of [Borel] and [Bourbaki] (k-rationality), [Gille-Szamuely] (tensors), [Jahnel] (Galois descent), [Serre] (Galois descent and torsors), and [Artin] (Severi-Brauer varieties).

Galois Descent and Severi-Brauer Varieties