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Mild pro-\(p\)-groups and Galois groups of \(p\)-extensions of \(\mathbb Q\). (English) Zbl 1122.11076

Let \(p\) be an odd prime and let \(G = F/R\) be a finitely presented pro-\(p\)-group, where \(F\) is the free pro-\(p\)-group of generator rank \(m\) and \(R\) the closed normal subgroup of \(F\) with minimal set of generators \(\{r_1,\ldots ,r_d\}.\) Let \(gr(G) = \bigoplus_{n=1}^{\infty}gr_n(G)\) be the graded Lie algebra associated to the \(p\)-central series \(\{G_n|n\in \mathbb N\}\) of \(G\). For \(r\in F\) the largest \(n\) with \(r\in F_n\) is called the filtration degree of \(r\) and \(\overline{r}\in gr_n(F)\) the initial form of \(r\). Let \(\rho_i\) be the initial form of \(r_i\) and let \({\mathfrak r}\) be the the ideal of \(L:= gr(F)\) generated by \(\rho_1,\ldots ,\rho_d\). Then \(L\) is a Lie algebra over \(\mathbb F_p[\pi]\), where \(\pi\) is induced by \(x\to x^p\).
The sequence \(\rho_1,\ldots ,\rho_d\) is called strongly free if the enveloping algebra \(U_{{\mathfrak g}}\) of \({\mathfrak g}:= L/{\mathfrak r}\) is a free \(\mathbb F_p[\pi]\)-module and \(M:={\mathfrak r}/[{\mathfrak r},{\mathfrak r}]\) is a free \(U_{{\mathfrak g}}\)-module on the images of \(\rho_1,\ldots ,\rho_d\) in \(M\). In this case one says that the presentation \(G=F/R\) is strongly free. \(G\) is called mild if it has a strongly free representation. This notion is due to D. Anick [J. Algebra 111, 154–165 (1987; Zbl 0635.20015)]. The author of the paper at hand proves among other results that a mild group has cohomological dimension \(2\).
Let \(S\) be a finite set of primes not containing \(p\) and let \(G_S(p)\) be the maximal \(p\)-extension of \(\mathbb Q\) unramified outside \(S\). Since a prime \(q\in S\) can be ramified only if \(q\equiv 1 \bmod p\) we assume \(q\equiv 1 \bmod p\) for \(q\in S\). Then \(G_S(p)\) is a pro-\(p\)-group of generator and relation rank \(|S|\) [I. R. Shafarevich, Am. Math. Soc., Transl., II. Ser. 59, 128–149 (1966; Zbl 0199.09707); translation from Publ. Math., Inst. Haut. Etud. Sci. 18 (1963), 295–319 (1964; Zbl 0118.27505)]. The structure of the relations is known up to third commutators [H. Koch, J. Reine Angew. Math. 219, 30–61 (1965; Zbl 0137.02402), see as well H. Koch, Galois theory of \(p\)-extensions, Springer (2002; Zbl 1023.11002)]. The structure of \(G_S(p)/G_S(p)^{(3)}\) was already determined by A. Fröhlich in his paper [“On fields of class two”, Proc. Lond. Math. Soc., III. Ser. 4, 235–256 (1954; Zbl 0055.03301)].
In the paper at hand the author shows that \(G_S(p)\) is a mild group if the primes of \(S\) satisfy a certain condition. E.g. this condition is satisfied if \(p=3\) and \(S=\{7,13,19,31\}\). Hence in this case \(G_S(3)\) is a pr-\(3\)-group of cohomological dimension \(2\) against the earlier belief of many mathematicians that the groups \(G_S(p)\) never have cohomological dimension 2 (if \(p\) is not in \(S\)). More general the author proves that for any finite set \(S\) of primes \(q\equiv 1\bmod p\) with \(|S|\geq 2\) there exists an extension of \(S\) to a set \(S'\) with \(|S'|= 2|S|\) such that \(G_{S'}(p)\) is a mild group.

MSC:

11R32 Galois theory
20E18 Limits, profinite groups
11S15 Ramification and extension theory
11S20 Galois theory
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