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

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.

scholarly journals Four-fold Massey products in Galois cohomology

2018 ◽  
Vol 154 (9) ◽  
pp. 1921-1959 ◽  
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.

scholarly journals Vanishing of Massey Products and Brauer Groups

2015 ◽  
Vol 58 (4) ◽  
pp. 730-740 ◽  
Author(s):  
Ido Efrat ◽  
Eliyahu Matzri
Keyword(s):  
Prime Number ◽  
Galois Cohomology ◽  
Brauer Groups ◽  
Central Simple Algebras ◽  
Massey Products ◽  
Global Fields ◽  
Root Of Unity ◽  
Simple Algebras ◽  
Central Simple

AbstractLet 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. We prove a stronger form of the vanishing property for global fields.

The Norm Residue Theorem in Motivic Cohomology

2019 ◽  
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Residue Theorem ◽  
Motivic Cohomology ◽  
Norm Residue

scholarly journals Right-angled Artin groups and enhanced Koszul properties

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Alberto Cassella ◽  
Claudio Quadrelli
Keyword(s):  
Finite Field ◽  
Prime Number ◽  
Galois Cohomology ◽  
Finite Graph ◽  
Galois Groups ◽  
Cohomology Algebra ◽  
Artin Group ◽  
Artin Groups ◽  
Koszul Algebra ◽  
Right Angled Artin 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.

A general reciprocity law for symbols on arbitrary vector spaces

2018 ◽  
Vol 17 (06) ◽  
pp. 1850101
Author(s):  
Fernando Pablos Romo
Keyword(s):  
General Theory ◽  
General Expression ◽  
Vector Spaces ◽  
Residue Theorem ◽  
Arbitrary Vector ◽  
The Past ◽  
Reciprocity Laws ◽  
Reciprocity Law ◽  
Norm Residue ◽  
Hilbert Norm

The aim of this work is to offer a general theory of reciprocity laws for symbols on arbitrary vector spaces and to show that classical explicit reciprocity laws are particular cases of this theory (sum of valuations on a complete curve, Residue Theorem, Weil Reciprocity Law and the Reciprocity Law for the Hilbert Norm Residue Symbol). Moreover, several reciprocity laws introduced over the past few years by D. V. Osipov, A. N. Parshin, I. Horozov, I. Horozov — M. Kerr and the author — together with D. Hernández Serrano — can also be deduced from this general expression.

The Norm Residue Theorem in Motivic Cohomology

2019 ◽  
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Residue Theorem ◽  
Motivic Cohomology ◽  
Norm Residue
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