Math. Z. https://doi.org/10.1007/s00209-018-2079-0 Mathematische Zeitschrift A generalization of Serre’s condition (F) with applications to the ﬁniteness of unramiﬁed cohomology Igor A. Rapinchuk Received: 12 August 2017 / Accepted: 14 February 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract In this paper, we introduce a condition (F )onaﬁeld K , for a positive integer m, that generalizes Serre’s condition (F) and which still implies the ﬁniteness of the Galois cohomology of ﬁnite Galois modules annihilated by m and algebraic K -tori that split over an extension of degree dividing m, as well as certain groups of étale and unramiﬁed cohomology. Various examples of ﬁelds satisfying (F ), including those that do not satisfy (F), are given. 1 Introduction In , Serre introduced the following condition on a proﬁnite group G: (F) For every integer m ≥ 1, G has ﬁnitely many open subgroups of index m. (see [30, Ch. III, Sect. 4.2]). He then deﬁned a perfect ﬁeld K to be of type (F) if the absolute Galois group G = Gal(K /K ) satisﬁes (F)—notice that this is equivalent to the fact that for every integer m, the (ﬁxed) separable closure K contains only ﬁnitely many
Mathematische Zeitschrift – Springer Journals
Published: May 29, 2018
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