On the behaviour of Brauer $p$-dimensions under finitely-generated field extensions

Abstract

The present paper shows that if $q\in \mathbb{P}$ or $q=0$, where $\mathbb{P}$ is the set of prime numbers, then there exist characteristic $q$ fields $E_{q,k}:k\in \mathbb{N}$, of Brauer dimension Brd$(E_{q,k})=k$ and infinite absolute Brauer $p$-dimensions abrd${}_p(E_{q,k})$, for all $p\in \mathbb{P}$ not dividing $q^2-q$. This ensures that Brd${}_p(F_{q,k})=\infty$, $p†q^2-q$, for every finitely-generated transcendental extension $F_{q,k}/E_{q,k}$. We also prove that each sequence $a_p,b_p$, $p\in \mathbb{P}$, satisfying the conditions $a_2=b_2$ and $0\le b_p\le a_p\le \infty$, equals the sequence abrd${}_p(E),{\mathrm{Brd}}_p(E)$, $p\in \mathbb{P}$, for a field $E$ of characteristic zero.

LaTeX, 14 pages: published in Journal of Algebra {\bf 428} (2015), 190-204; the abstract in the Metadata updated to fit the one of the paper

References in corpus (4)

• Applications of patching to quadratic forms and central simple algebras
• Period-index and u-invariant questions for function fields over complete discretely valued fields
• On Brauer $p$-dimensions and index-exponent relations over finitely-generated field extensions
• Demushkin groups and inverse Galois theory for pro-p-groups of finite rank and maximal p-extensions

Cited by in corpus (4)

• On Brauer $p$-dimensions and index-exponent relations over finitely-generated field extensions
• On index-exponent relations over Henselian fields with local residue fields
• On the Brauer $p$-dimension of Henselian discrete valued fields of residual characteristic $p>0$
• Lower bounds and infinity criterion for Brauer $p$-dimensions of finitely-generated field extensions