Triple Massey products of weight \((1,n,1)\) in Galois cohomology
type: Item
instance of: scholarly article
title: Triple Massey products of weight \((1,n,1)\) in Galois cohomology
zbMATH Open document ID: 1425.12005
author: Eliyahu Matzri
published in: Journal of Algebra
publication date: 2018-03-01T00:00:00Z
review text: Let \(F\) be a field containing a primitive \(p\)-th root of unity, for an arbitrary fixed prime number \(p\). Denote by \(\mathcal{G}_F\) the absolute Galois group of \(F\), and by \(H^n\) the modulo \(p\) (continuous) cohomology group \(H^n(\mathcal{G}_F, \mathbb Z/p\mathbb Z)\). Given an integer \(d \ge 2\), the \(d\)-fold Massey product is a family, indexed by \(\mathbb N^d\), of certain semi-defined, multi-valued functions which we denote by dMP of weight \((n_1, \ldots, n_d)\) (like the author, we skip the precise definition but refer the reader for it to the paper by \textit{D. Kraines} [Trans. Am. Math. Soc. 124, 431--449 (1966; Zbl 0146.19201)]. In particular, for \(d = 3\), the 3MP of weight \((n, k, m)\) is a function as above, \(\langle ., . , . \rangle: H^n \times H^k \times H^m \to H^{n+k+m-1}\). A defined dMP is called essential if it does not contain the zero element. A conjecture stated by \textit{J. Mináč} and \textit{N. D. Tân} [J. Eur. Math. Soc. (JEMS) 19, No. 1, 255--284 (2017; Zbl 1372.12004)] predicts that, for absolute Galois groups and \(d \ge 3\), there are no essential dMP of weight \((1, 1, \ldots, 1)\). In recent years, this conjecture has attracted the attention of a number of algebraists (including the author among the major contributors), and its investigation tends to become one of the main directions of the research on the problem of characterizing absolute Galois groups of fields in the class of profinite groups. The conjecture has first been proved in the author's preprint available at [\url{arXiv:1411.4146}]; the proof relies essentially on the theory of (finite-dimensional associative) central simple algebras, specifically, on properties of abelian crossed products. More recently, the same result has been obtained with Galois cohomological proofs by \textit{J. Mináč} and \textit{N. D. Tân} [J. Lond. Math. Soc., II. Ser. 94, No. 3, 909--932 (2016; Zbl 1378.12002)], and \textit{I. Efrat} and the author [J. Eur. Math. Soc. (JEMS) …
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 16K50
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 55S30
zbMATH DE Number: 6845886
zbMATH Keywords: Massey products
zbMATH Keywords: weight of a Massey product
zbMATH Keywords: Galois cohomology
zbMATH Keywords: external cohomological operation
zbMATH Keywords: absolute Galois group
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2017.11.036
OpenAlex ID: W2774248079
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DOI: 10.1016/J.JALGEBRA.2017.11.036
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description: scientific article; zbMATH DE number 6845886