Profinite Groups with a Cyclotomic pp-Orientation

  • Claudio Quadrelli

    Department of Mathematics and Applications, Università di Milano-Bicocca, Via R. Cozzi 55 - ed. U5, 20125 Milan, Italy
  • Thomas S. Weigel

    Department of Mathematics and Applications, Università di Milano-Bicocca, Via R. Cozzi 55 - ed. U5, 20125 Milan, Italy
Profinite Groups with a Cyclotomic $p$-Orientation cover
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Abstract

Let pp be a prime. A continuous representation θ ⁣:GGL1(Zp)\theta\colon G\to\text{GL}_1(\mathbb{Z}_p) of a profinite group GG is called a cyclotomic pp-orientation if for all open subgroups UGU\subseteq G and for all k,n1k,n\geq1 the natural maps Hk(U,Zp(k)/pn)Hk(U,Zp(k)/p)H^k(U,\mathbb{Z}_p(k)/p^n)\to H^k(U,\mathbb{Z}_p(k)/p) are surjective. Here Zp(k)\mathbb{Z}_p(k) denotes the Zp\mathbb{Z}_p-module of rank 1 with UU-action induced by θUk\theta\vert_U^k. By the Rost-Voevodsky theorem, the cyclotomic character of the absolute Galois group GKG_{\mathbb{K}} of a field K\mathbb{K} is, indeed, a cyclotomic pp-orientation of GKG_{\mathbb{K}}. We study profinite groups with a cyclotomic pp-orientation. In particular, we show that cyclotomicity is preserved by several operations on profinite groups, and that Bloch-Kato pro-pp groups with a cyclotomic pp-orientation satisfy a strong form of Tits' alternative and decompose as semi-direct product over a canonical abelian closed normal subgroup.

Cite this article

Claudio Quadrelli, Thomas S. Weigel, Profinite Groups with a Cyclotomic pp-Orientation. Doc. Math. 25 (2020), pp. 1881–1916

DOI 10.4171/DM/788