Function fields of one variable over PAC fields
From the introduction: Conjecture II of Serre considers a field \(F\) of characteristic \(p\) with \(\text{cd}(\text{Gal}(F))\leq 2\) such that either \(p = 0\) or \(p > 0\) and \([F:F^p] \leq p\) and predicts that \(H^1(\text{Gal}(F),G) = 1\) (i.e. each principal homogeneous \(G\)-spaces has an \(F\)-rational point) for each simply connected semi-simple linear algebraic group \(G\) [\textit{J.-P. Serre}, Galois cohomology. Berlin: Springer (1997; Zbl 0902.12004), p. 139]. As Serre notes, the hypothesis of the conjecture holds in the case where \(F\) is a field of transcendence degree 1 over a perfect field \(K\) with \(\text{cd}(\text{Gal}(K))\leq 1\). Indeed, in this case \(\text{cd}(\text{Gal}(F))\leq 2\) [Serre (loc.cit.), p. 83, Prop. 11] and \([F:F^p] \leq p\) if \(p > 0\) (by the theory of \(p\)-bases [\textit{M. D. Fried} and \textit{M. Jarden}, Field arithmetic. 3rd rev. ed. Berlin: Springer (2008; Zbl 1145.12001), Lemma 2.7.2]). We prove the conjecture for \(F\) in the special case, where \(K\) is PAC of characteristic 0 that contains all roots of unity. One of the main ingredients of the proof is the projectivity of \(\text{Gal}(K(x)_{\text{ab}})\) (where \(x\) is transcendental over \(K\) and \(K(x)_{\text{ab}}\) is the maximal Abelian extension of \(K(x)\)). We also use the same ingredient to establish an analog to the well-known open problem of Shafarevich that \(\text{Gal}(\mathbb Q_{\text{ab}})\) is free. Under the assumption that \(K\) is PAC and contains all roots of unity we prove that \(\text{Gal}(K(x)_{\text{ab}})\) is not only projective but even free. This proves a stronger version of a conjecture of Bogomolov for a function field of one variable \(F\) over a PAC field that contains all roots of unity [\textit{L. Positselski}, Int. Math. Res. Not. 2005, No. 31, 1901--1936 (2005; Zbl 1160.19301), Conjecture 1.1].
scientific article; zbMATH DE number 5663786