3-fold Massey products in Galois cohomology -- The non-prime case

Abstract

For m ≥ 2, let F be a field of characteristic prime to m and containing the roots of unity of order m, and let G F be its absolute Galois group. We show that the 3-fold Massey products χ 1 , χ 2 , χ 3 , with χ 1 , χ 2 , χ 3 ∈ H 1 (G F , Z/m) and χ 1 , χ 3 Z/m-linearly independent, are non-essential. This was earlier proved for m prime. Our proof is based on the study of unitriangular representations of G F .A major open problem in modern Galois theory is to characterize the profinite groups which are realizable as the absolute Galois group G F = Gal(F sep /F ) of a field F . Here it is natural to study the cohomological structure of absolute Galois groups, and in particular, the cohomology ring H * (G F , Z/m), under the standard assumption that F has characteristic not dividing m and contains the roots of unity of order m. The celebrated Voevodsky-Rost theorem fully describes H * (G F , Z/m) as a graded algebra with respect to the cup product. Namely, it is the quotient of the tensor Z/m-algebra over H 1 (G F , Z/m) modulo the ideal generated by all tensors (a) F ⊗ (1 − a) F , where 0, 1 = a ∈ F , and (•) F denotes Kummer elements. This imposes strong restrictions on the group-theoretic structure of G F ([CEM12], [EfMi17]).In recent years there has been extensive research regarding external cohomological operations on H * (G F , Z/m), that is, natural maps on this graded algebra defined using the cochain algebra C * (G F , Z/m), and which go beyond the ring structure. A major example is the n-fold Massey product •, . . . ,where n ≥ 2. This operation can be defined in the cohomology algebra of any differential graded algebraand is a multi-valued map, i.e., for cohomology classes [a 1 ], . . . , [a n ] ∈ H 1 (A • ), the Massey product [a 1 ], . . . , [a n ] is a subset of H 2 (A • ). For n = 2 one has [a 1 ], [a 2 ] = {[a 1 ][a 2 ]}, so in our case the 2-fold Massey product just recovers the cup product.