Lower bounds and infinity criterion for Brauer $p$-dimensions of finitely-generated field extensions

Abstract

Let $E$ be a field, $p$ a prime number and $F/E$ a finitely-generated extension of transcendency degree $t$. This paper shows that if the absolute Galois group ${\mathcal{G}}_E$ is of nonzero cohomological $p$-dimension cd${}_p(E)$, then the field $F$ has Brauer $p$-dimension Brd${}_p(F)\ge t$ except, possibly, in case $p=2$, the Sylow pro-2-subgroups of ${\mathcal{G}}_E$ are of order 2, and $F$ is a nonreal field. It announces that Brd${}_p(F)$ is infinite whenever $t\ge 1$ and the absolute Brauer $p$-dimension abrd${}_p(E)$ is infinite; moreover, for each pair $(m,n)$ of integers with $1\le m\le n$, there exists a central division $F$-algebra of exponent $p^m$ and Schur index $p^n$.

8 pages, LaTeX. Published in C.R. Acad. Bulg. Sci. 66 (2013), No. 7, 923-932. Paper 15 (by Parimala and Suresh in the References) has been published in Invent. Math. 197 (2014), 215-235. An abridged version of paper 3 will appear in J. Algebra 428 (2015), 190-204; for the rest of the material in paper 3, see arXiv:1501.05977v1 [math.RA], Jan. 23, 2015

References in corpus (2)

• Period-index and u-invariant questions for function fields over complete discretely valued fields
• On the behaviour of Brauer $p$-dimensions under finitely-generated field extensions