For a prime number ℓ we say that an oriented pro-ℓ group (G, θ) has the Bogomolov–Positselski property if the kernel of the canonical projection on its maximal θ-abelian quotient πG,θab:G→G(θ) is a free pro-ℓ group contained in the Frattini subgroup of G. We show that oriented pro-ℓ groups of elementary type have the Bogomolov–Positselski property (cf. Theorem 1.2). This shows that Efrat’s Elementary Type Conjecture implies a positive answer to Positselski’s version of Bogomolov’s Conjecture on maximal pro-ℓ Galois groups of a field K in case that K×/(K×)ℓ is finite. Secondly, it is shown that for an H∙-quadratic oriented pro-ℓ group (G, θ) the Bogomolov–Positselski property can be expressed by the injectivity of the transgression map d22,1 in the Hochschild–Serre spectral sequence (cf. Theorem 1.4).

Quadrelli, C., Weigel, T. (2022). Oriented pro-ℓ groups with the Bogomolov-Positselski property. RESEARCH IN NUMBER THEORY, 8(2) [10.1007/s40993-022-00318-9].

Oriented pro-ℓ groups with the Bogomolov-Positselski property

Quadrelli, C
;
Weigel, T.
2022

Abstract

For a prime number ℓ we say that an oriented pro-ℓ group (G, θ) has the Bogomolov–Positselski property if the kernel of the canonical projection on its maximal θ-abelian quotient πG,θab:G→G(θ) is a free pro-ℓ group contained in the Frattini subgroup of G. We show that oriented pro-ℓ groups of elementary type have the Bogomolov–Positselski property (cf. Theorem 1.2). This shows that Efrat’s Elementary Type Conjecture implies a positive answer to Positselski’s version of Bogomolov’s Conjecture on maximal pro-ℓ Galois groups of a field K in case that K×/(K×)ℓ is finite. Secondly, it is shown that for an H∙-quadratic oriented pro-ℓ group (G, θ) the Bogomolov–Positselski property can be expressed by the injectivity of the transgression map d22,1 in the Hochschild–Serre spectral sequence (cf. Theorem 1.4).
Articolo in rivista - Articolo scientifico
Bogomolov’s Conjecture; Kummerian oriented pro-ℓ groups; Maximal pro-ℓ Galois groups; Oriented pro-ℓ groups;
English
22
Quadrelli, C., Weigel, T. (2022). Oriented pro-ℓ groups with the Bogomolov-Positselski property. RESEARCH IN NUMBER THEORY, 8(2) [10.1007/s40993-022-00318-9].
Quadrelli, C; Weigel, T
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/351878
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