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

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International Electronic Journal of Algebra
Volume:2 Issue:2
DIVISION ALGEBRAS THAT RAMIFY ONLY ON THE ZEROS OF AN ELEMENTARY SYMMETRIC POLYNOMIAL

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

Authors : Timothy J Ford

Pages : 189-207

View : 13 | Download : 9

Publication Date : 2007-12-01

Article Type : Research Paper

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

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