Four-fold Massey products in Galois cohomology

Guillot, Pierre; Mináč, Ján; Topaz, Adam

doi:10.1112/s0010437x18007297

Cited by 19 publications

(11 citation statements)

References 43 publications

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“…On the other hand, Harpaz and Wittenberg show in [HaW19] that the above property of 3-fold Massey products does not hold for F = Q and m = 8, so the assumption about roots of unity cannot be removed in general. Following [GMTW18], [GM19], they prove the above fact for general n-fold Massey products, n ≥ 3, in the case where F is a number field and m = p is prime. We refer to [PS16] for related results.…”

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confidence: 89%

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

Efrat

2020

Preprint

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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.

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confidence: 89%

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

Efrat

2020

Preprint

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“…[34]) it can be shown that the non-trivial element of B ω (V )/Br 0 (V ) is in fact locally constant and yields a nontrivial obstruction to the Hasse principle (see [GMT18,Example A.15]). We note that this is by no means a contradiction to Theorem 5.1: indeed, an obstruction coming from a locally constant Brauer class means that the homomorphism α : Γ k −→ A = U/U 1 does not lift to U/U 3 = U/Z, and hence the relevant Massey product is not defined.…”

Section: More Brauer Group Computationsmentioning

confidence: 99%

“…The claim in the general non-trivial case was proven in [MT17b, Theorem 4.3], using local duality. (3) When k is a number field, n = 4 and p = 2 [GMT18].…”

Section: Introductionmentioning

confidence: 99%

The Massey vanishing conjecture for number fields

Harpaz¹,

Wittenberg²

2019

Preprint

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“…But how does one look for those properties that distinguish absolute p-Galois groups from the broader class of pro-p groups? There are several research agendas that aim to answer this question using a number of different techniques, including the study of Massey products (see [39,50,60,84,92,93,94,117,118,120]) and the Koszulity of Galois cohomology (see [82,83,102]).…”

Section: Introductionmentioning

confidence: 99%

Galois module structure of square power classes for biquadratic extensions

Chemotti,

Minac,

Schultz

et al. 2021

Preprint

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This paper marks the first time that the Galois module structure of power classes of a field has been fully determined when the modular representation theory allows for an infinite number of indecomposable types. The example and methodology we consider pave the way for a paradigm shift in the study of absolute Galois groups via Galois modules. Whereas generic modules of this type are completely intractable to approach, this work shows that Galois cohomology provides a context where this otherwise impossible problem becomes approachable, interesting, and illuminating.Let V = Z/2Z⊕Z/2Z. For a Galois extension K/F with char(K) = 2 and Gal(K/F ) V , we determine the F 2 [Gal(K/F )]-module structure of K × /K ×2 . Although there are an infinite number of (pairwise non-isomorphic) indecomposable F 2 [V ]-modules, our decomposition includes at most 8 indecomposable types. This is a dramatic development on several fronts. Most conspicuously, it shows that these modules can be computed in settings which might initially seem too forbidding. The elegant nature of this decomposition also points to new structural restrictions on absolute Galois groups. More generally, the techniques used in this paper provide a blueprint for studying a broad class of Galois modules. This opens the door for a novel, systematic study of the Galois theory of maximal p-extensions of fields.Prepare for the unknown by studying how others in the past have coped with the unforeseeable and the unpredictable.

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Four-fold Massey products in Galois cohomology

Cited by 19 publications

References 43 publications

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

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

The Massey vanishing conjecture for number fields

Galois module structure of square power classes for biquadratic extensions

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