DIVISION ALGEBRAS THAT RAMIFY ONLY ON THE ZEROS OF AN ELEMENTARY SYMMETRIC POLYNOMIAL

Year 2007, Volume: 2 Issue: 2, 189 - 207, 01.12.2007

Timothy J. Ford

Abstract

Let k be an algebraically closed field of characteristic zero. The elementary symmetric polynomial of degree n − 1 in n variables is a homogeneous polynomial, hence defines both an affine variety in An k which we denote by Cn−1 and a projective variety in Pn−1k denoted Vn−1. We describe, up to Brauer equivalence, the central division algebras over the rational function field of An which ramify only on Cn−1 as well as the central division algebras over the rational function field of Pn−1 that ramify only on Vn−1. The Brauer group of the cubic surface V3 in P3 is computed and is shown to consist solely of Azumaya algebras that are locally trivial in the Zariski topology.

Keywords

Brauer group, division algebra, elementary symmetric polynomial