Triple Massey products of weight \((1,n,1)\) in Galois cohomology
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) 19, No. 12, 3629--3640 (2017; Zbl 1425.12004)]; furthermore, the former two authors have established the result, for an arbitrary ground field (not necessarily containing a primitive \(p\)-th root of unity). Restricting to ground fields \(F\) containing a primitive \(p\)-th root of unity, the paper under review generalizes the above-noted result in two directions. First, it proves, for a prime \(p > 2\) and an odd positive integer \(n\), that if any defined 3MP of weight \((n, m, n)\) of the form \(\langle \alpha , \beta , \alpha \rangle \) is inessential (i.e. it contains zero), then so is any defined 3MP of weight \((n, m, n)\). In particular, this implies that any defined 3MP of weight \((1, n, 1)\) is inessential.
scientific article; zbMATH DE number 6845886
scientific article