On Brauer \(p\)-dimensions and absolute Brauer \(p\)-dimensions of Henselian fields
Let \(E\) be a field. For each prime number \(p\), let \(Brd_p(E)\) be the Brauer \(p\)-dimension of \(E\) and \(abrd_p(E)\) be the absolute Brauer \(p\)-dimension of \(E\). In this paper, the author studies the sequences \(Brd_p(E)\) and \(abrd_p(E)\), when \(p\) runs over the set \(\mathbb{P}\) of the prime numbers. Suppose that \((E,v)\) is a Henselian valued field with residue field \(\hat E\). For \(p\in\mathbb{P}\), under some restrictions on \(\hat E\), such as \(abrd_p(\hat E)=0\), the author determines \(Brd_p(E)\) and \(abrd_p(E)\) if \(\mathrm{char}(\hat E)\not=p\). Let \(\Sigma_0\) be the set of sequences \(Brd_p(E)\) and \(abrd_p(E)\) where \(p\in \mathbb{P}\) and \(E\) runs across the class of Henselian valued fields with \(\mathrm{char}(\hat E)\not=0\) and a projective absolute Galois group \(\mathcal G_{\hat E}\). The author gives a description for \(\Sigma_0\). Especially, \(\Sigma_0\) admits a sequence \(a_p,b_p\in\mathbb{N}\cup\{0,\infty\}\), \(p\in \mathbb{P}\), with \(a_2\leq 2b_2\) and \(a_p\geq b_p\). The paper covers similar results in the case of nonzero characteristic.
scientific article