On Galois cohomology and realizability of 2-groups as Galois groups

AbstractLet 𝔽 be a finite field. We prove that the cohomology algebra H^{\bullet}(G_{\Gamma},\mathbb{F}) with coefficients in 𝔽 of a right-angled Artin group G_{\Gamma} is a strongly Koszul algebra for every finite graph Γ. Moreover, H^{\bullet}(G_{\Gamma},\mathbb{F}) is a universally Koszul algebra if, and only if, the graph Γ associated to the group G_{\Gamma} has the diagonal property. From this, we obtain several new examples of pro-𝑝 groups, for a prime number 𝑝, whose continuous cochain cohomology algebra with coefficients in the field of 𝑝 elements is strongly and universally (or strongly and non-universally) Koszul. This provides new support to a conjecture on Galois cohomology of maximal pro-𝑝 Galois groups of fields formulated by J. Mináč et al.

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Two families of pro-𝑝 groups that are not absolute Galois groups

Journal of Group Theory ◽

10.1515/jgth-2020-0186 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Claudio Quadrelli

Keyword(s):

Galois Cohomology ◽

Galois Groups ◽

Residue Theorem ◽

Smoothness Property ◽

Massey Products ◽

Norm Residue

Abstract Let 𝑝 be a prime. We produce two new families of pro-𝑝 groups which are not realizable as absolute Galois groups of fields. To prove this, we use the 1-smoothness property of absolute Galois pro-𝑝 groups. Moreover, we show in these families, one has several pro-𝑝 groups which may not be ruled out as absolute Galois groups employing the quadraticity of Galois cohomology (a consequence of the norm residue theorem), or the vanishing of Massey products in Galois cohomology.

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On Galois cohomology and realizability of 2-groups as Galois groups II

Central European Journal of Mathematics ◽

10.2478/s11533-011-0086-z ◽

2011 ◽

Vol 9 (6) ◽

pp. 1333-1343 ◽

Cited By ~ 1

Author(s):

Ivo M. Michailov

Keyword(s):

Galois Cohomology ◽

Galois Groups

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COHOMOLOGY AND OVERCONVERGENCE FOR REPRESENTATIONS OF POWERS OF GALOIS GROUPS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748019000197 ◽

2019 ◽

pp. 1-61

Author(s):

Aprameyo Pal ◽

Gergely Zábrádi

Keyword(s):

Galois Cohomology ◽

Galois Groups ◽

Cohomology Groups ◽

Direct Power ◽

Equivalence Of Categories

We show that the Galois cohomology groups of $p$ -adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$ can be computed via the generalization of Herr’s complex to multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules. Using Tate duality and a pairing for multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules we extend this to analogues of the Iwasawa cohomology. We show that all $p$ -adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$ are overconvergent and, moreover, passing to overconvergent multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules is an equivalence of categories. Finally, we prove that the overconvergent Herr complex also computes the Galois cohomology groups.

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Four-fold Massey products in Galois cohomology

Compositio Mathematica ◽

10.1112/s0010437x18007297 ◽

2018 ◽

Vol 154 (9) ◽

pp. 1921-1959 ◽

Cited By ~ 1

Author(s):

Pierre Guillot ◽

Ján Mináč ◽

Adam Topaz

Keyword(s):

Number Field ◽

Galois Cohomology ◽

Number Fields ◽

Galois Groups ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Massey Products ◽

Necessary And Sufficient ◽

Global Principle

In this paper, we develop a new necessary and sufficient condition for the vanishing of $4$-Massey products of elements in the modulo-$2$ Galois cohomology of a field. This new description allows us to define a splitting variety for $4$-Massey products, which is shown in the appendix to satisfy a local-to-global principle over number fields. As a consequence, we prove that, for a number field, all such $4$-Massey products vanish whenever they are defined. This provides new explicit restrictions on the structure of absolute Galois groups of number fields.

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