On Brauer $p$-dimensions and index-exponent relations over finitely-generated field extensions

Abstract

Let $E$ be a field of absolute Brauer dimension abrd$(E)$, and $F/E$ a transcendental finitely-generated extension. This paper shows that the Brauer dimension Brd$(F)$ is infinite, if abrd$(E)=\infty$. When the absolute Brauer $p$-dimension abrd${}_p(E)$ is infinite, for some prime number $p$, it proves that for each pair $(n,m)$ of integers with $n\ge m>0$, there is a central division $F$-algebra of Schur index $p^n$ and exponent $p^m$. Lower bounds on the Brauer $p$-dimension Brd${}_p(F)$ are obtained in some important special cases where abrd${}_p(E)<\infty$. These results solve negatively a problem posed by Auel et al. (Transf. Groups {\bf 16}: 219-264, 2011).

19 pages, LaTeX; "coordinates" of three references updated; to appear in Manuscripta Mathematica. arXiv admin note: substantial text overlap with arXiv:1207.0965

References in corpus (1)

• Applications of patching to quadratic forms and central simple algebras