The symbol length for elementary type pro-\(p\) groups and Massey products (Q6558486)

Let \(p\) be a fixed prime number, and for any profinite group \(G\) and an integer \(n \ge 0\), denote by \(H^n(G, \mathbb{F}_p)\) the \(n\)-th profinite cohomology group of \(G\) with respect to its trivial action on \(\mathbb{F}_p\); also, let \(H^*(G, \mathbb{F}_p) = \bigoplus_{n \ge 0} H^n(G, \mathbb{F}_p)\) be the cohomology (graded) ring with the cup product. A symbol in \(H^n(G, \mathbb{F}_p)\), where \(n \ge 1\), is a cup product \(\chi_1 \cup \dots \chi_n\) of \(n\) elements \(\chi_1, \dots, \chi_n\) of \(H^1(G_F, \mathbb{F}_p)\). The symbol length sl\((\omega)\) of a cohomology class \(\omega \in H^n(G, \mathbb{F}_p)\) is the minimal positive integer \(m\) such that \(\omega\) can be written as a sum of \(m\) symbols in \(H^n(G, \mathbb{F}_p)\); if no such integer exists, one sets sl\((\omega) = \infty\).\N\NLet now \(F\) be a field, \(F^{\ast} = F \setminus \{0\}\) its multiplicative group, \(F_{sep}\) a separable closure of \(F\), and \(G_F = Gal(F^{sep}/F)\) the absolute Galois group of \(F\); suppose that \(F\) contains a primitive \(p\)-th root of unity. By the Norm Residue Theorem (the Bloch-Kato conjecture proved by Voevodsky and Rost), \(H^{\ast}(G_F, \mathbb{F}_p)\) is isomorphic as a graded ring to the mod-\(p\) Milnor \(K\)-ring \(K_{\ast}^M(F)/p\) (see [\textit{V. Voevodsky}, Ann. Math. (2) 174, No. 1, 401--438 (2011; Zbl 1236.14026); \textit{C. Haesemeyer} and \textit{C. A. Weibel}, The norm residue theorem in motivic cohomology. Princeton, NJ: Princeton University Press (2019; Zbl 1433.14001)]). Consequently, \(H^{\ast}(G_F, \mathbb{F}_p)\) is generated by its degree \(1\) elements, that is, sl\((\omega) < \infty\) for all \(\omega\). General bounds on sl\((\omega)\) can serve as a measure for the arithmetical complexity of \(F\), as well as of specific cohomological constructions (see e.g., [\textit{D. Krashen}, Bull. Lond. Math. Soc. 48, No. 6, 985--1000 (2016; Zbl 1405.11039)], and [\textit{E. Matzri}, Trans. Am. Math. Soc. 368, No. 1, 413--427 (2016; Zbl 1339.12002)]). However, the proof of the Norm Residue Theorem is not constructive, and does not yield such explicit bounds.\N\NThis attracts interest in finding explicit bounds for the symbol length of classes of elements of \(H^n(G_F, \mathbb{F}_p)\) which are pullbacks of a given cohomology class in a quotient of \(G_F\). A major problem motivating the research in this direction is Leonid Positselski's conjecture, stated as follows:\N\NConjecture 1.1. Given a pro-\(p\) group \(\overline P\), an integer \(n > 0\), and \(\bar \omega \in H^n(\overline G, \mathbb{F}_p)\), there is a non-negative integer \(M = M(\overline G, n, \bar \omega)\) such that for every field \(F\) containing a primitive \(p\)-th root of unity, and for every profinite group homomorphism \(\rho : G_F \to \overline G\), one has sl\((\rho^{\ast}(\bar \omega)) \le M\), where \(\rho^{\ast}: H^n(\overline G, \mathbb{F}_p) \to H^n(G_F, \mathbb{F}_p)\) is the induced (pullback) homomorphism.\N\NAs explained in the paper, the cohomology graded rings \(H^*(G_F, \mathbb{F}_p)\) and \(H^*(G_F(p), \mathbb{F}_p)\) can be identified, which allows to replace in Conjecture 1.1 \(G_F\) by its maximal pro-\(p\) quotient \(G_F(p)\), namely, the Galois group \(\mathcal{G}(F(p)/F)\), where \(F(p)\) is the maximal \(p\)-extension of \(F\) in \(F_{sep}\). In this setting, the first main result of the paper under review proves Conjecture 1.1, under the hypothesis that \(G_F(p)\) is a pro-\(p\) group of elementary type; when \(\overline G\) is a finite \(p\)-group, such a uniform bound is given explicitly. The hypothesis ensures that \(G_F(p)\) is a finitely-generated pro-\(p\) group, and the pro-\(p\) Elementary Type Conjecture (ETC) formulated by the author [in: Séminaire de Structures Algébriques Ordonnés, Lecture Notes, vol. 54, Univ. Paris VII (1995)] predicts that \(G_F(p)\) is of elementary type whenever it is a finitely-generated pro-\(p\) group. In addition, it is known (see Remark 3.4 in: [\textit{I. Efrat}, Manuscr. Math. 95, No. 2, 237--249 (1998; Zbl 0902.12003)]) that every pro-\(p\) group of elementary type is realizable as a maximal pro-\(p\) Galois group of a field containing a primitive \(p\)-th root of unity.\N\NThe second main result of the reviewed paper is obtained as an application of the first one. It states that if \(G\) is a pro-\(p\) group of elementary type and \(m \ge 2\) is an integer, then the elements of every \(m\)-fold Massey product in \(H^2(G, \mathbb{F}_p)\) have symbol length at most \(\lfloor m^2/4\rfloor + m\). When \(m \ge 3\), this result has also been obtained by \textit{C. Quadrelli} in [J. Number Theory 258, 40--65 (2024; Zbl 1537.12006)]; Corollary~1.3 of Quadrelli's paper gives an account of classes of fields for which ETC has been proved. As noted by the author, several other major conjectures on the structure of absolute and maximal pro-p Galois groups have recently proved for the class of pro-\(p\) groups of elementary type: these include Positselski's conjecture that \(H^*(G_F, \mathbb{F}_p)\) is a Koszul algebra (see [\textit{J. Mináč} et al., Adv. Math. 380, Article ID 107569, 50 p. (2021; Zbl 1483.12003)], and the Bogomolov-Positselski freeness conjecture concerning the maximal pro-\(p\) Galois group of the maximal radical extension of \(F\) (see [\textit{C. Quadrelli} and \textit{T. S. Weigel}, Res. Number Theory 8, No. 2, Paper No. 21, 22 p. (2022; Zbl 1492.12003)]), and other related results.