[Paper Review] The strong Massey vanishing conjecture for fields with virtual cohomological dimension at most $1$

Ambrus Pál, Endre Szabó|arXiv (Cornell University)|Nov 15, 2018

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper proves the strong Massey vanishing conjecture for fields with virtual cohomological dimension at most 1 and for pseudo-p-adically closed (PpC) fields, using results from Haran and Haran–Jarden on Galois cohomology and embedding problems. It further constructs a pro-2 group that satisfies weak Massey vanishing for n ≥ 3 but fails strong Massey vanishing at n = 4, demonstrating a strict hierarchy between the two properties.

ABSTRACT

We show that a strong vanishing conjecture for $n$-fold Massey products holds for fields of virtual cohomological dimension at most $1$ using a theorem of Haran. We also prove the same for PpC fields, using results of Haran--Jarden. Finally we construct a pro-$2$ group which satisfies the weak Massey vanishing property for every $n\geq3$, but does not satisfy the strong Massey vanishing property for $n=4$.

Motivation & Objective

• To establish the strong Massey vanishing conjecture for fields with virtual cohomological dimension at most 1.
• To extend the strong Massey vanishing property to pseudo-p-adically closed (PpC) fields.
• To clarify the distinction between weak and strong Massey vanishing by constructing a counterexample in the pro-2 setting.
• To demonstrate that the strong Massey vanishing property is strictly stronger than the weak one for n ≥ 4.

Proposed method

• Leverages a theorem of Haran on the structure of Galois groups of fields with virtual cohomological dimension ≤ 1 to prove strong Massey vanishing.
• Applies results from Haran–Jarden on PpC fields to show that their absolute Galois groups satisfy the strong Massey vanishing property.
• Uses Dwyer’s theorem on obstruction classes in embedding problems to analyze the vanishing of n-fold Massey products.
• Constructs a pro-2 group via inverse limits of Galois groups of number fields, using central extensions and obstruction theory.
• Employs the inflation-restriction exact sequence and cohomological techniques to analyze the kernel of pullback maps in group cohomology.
• Applies the naturality of obstruction classes to compare solutions of embedding problems across group homomorphisms and quotients.

Experimental results

Research questions

• RQ1 Does the strong Massey vanishing conjecture hold for fields with virtual cohomological dimension at most 1?
• RQ2 Do pseudo-p-adically closed fields satisfy the strong Massey vanishing property for all primes?
• RQ3 Is there a pro-2 group that satisfies weak Massey vanishing for all n ≥ 3 but fails strong Massey vanishing at n = 4?
• RQ4 What is the precise relationship between weak and strong Massey vanishing in the context of profinite groups?

Key findings

• The strong Massey vanishing conjecture holds for all fields K with cd(K(i)) ≤ 1, for every prime p.
• The strong Massey vanishing conjecture is valid for all pseudo-p-adically closed (PpC) fields, with respect to every prime number.
• A pro-2 group G is constructed that satisfies the weak Massey vanishing property for all n ≥ 3, but fails to satisfy the strong Massey vanishing property at n = 4.
• The obstruction class o(F₄(ψ̄⋆β)) is non-zero and lies in the kernel of τ*, which is at most one-dimensional, enabling comparison of solutions across group extensions.
• The n-fold Massey product ⟨κ₅¹∘ρₖ, κ₅²∘ρₖ, κ₅³∘ρₖ⟩ contains zero, implying the same for the original product in the group G.
• The construction shows that the strong Massey vanishing property is strictly stronger than the weak one for n = 4, as the weak condition holds but the strong does not.