Vanishing of Massey Products and Brauer Groups

Abstract

Abstract. Let p be a prime number and F a field containing a root of unity of order p. We relate recent results on vanishing of triple Massey products in the mod-p Galois cohomology of F, due to Hopkins, Wickelgren, Mináč, and Tân, to classical results in the theory of central simple algebras. For global fields, we prove a stronger form of the vanishing property.

Triple Massey products and absolute Galois groups

“…Moreover, it is conjectured in [MT15] that the n-fold Massey product above is never essential for every n ≥ 3. Also, in [EM15] we find close connections between these results and classical facts in the theory of central simple algebras. In particular, 2) is closely related to Albert's characterization from 1939 [Alb39] (as refined by Tignol [Tig79]; see also Rowen [Row84] and [Tig81]) of the central simple algebras of exponent 2 and degree 4 as biquaternionic algebras.…”

Triple Massey products vanish over all fields

“…In [19], it was proved that G F has the vanishing triple Massey product property with respect to F p for any global field F containing a primitive pth root of unity. In [8], Efrat and Matzri provided alternative proofs for the above-mentioned results in [17,19]. In [14], Matzri proved that, for any prime p and for any field F containing a primitive pth root of unity, G F has the vanishing triple Massey product property.…”

Four-fold Massey products in Galois cohomology

“…The investigation of Massey products in Galois cohomology of arbitrary fields has recently started progressing very rapidly. This surge started with the work of Hopkins and Wickelgren [HW15], and further progressed by Mináč and Tân [MT17b, MT17c, MT16, MT17a] and Efrat and Matzri [EM15, EM17, Efr14, Mat11]. In fact, ideas related to vanishing of modulo- triple Massey products already appeared in 2003 by Gao, Leep, Mináč and Smith [GLMS03], albeit using different terminology.…”