Mathematische Zeitschrift

, Volume 291, Issue 1–2, pp 199–213 | Cite as

A generalization of Serre’s condition \(\mathrm {(F)}\) with applications to the finiteness of unramified cohomology

  • Igor A. RapinchukEmail author


In this paper, we introduce a condition (\(\mathrm {F}_m'\)) on a field 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 (\(\mathrm {F}_m'\)), including those that do not satisfy (F), are given.



The author would like to thank the anonymous referee for a careful reading of the paper and A. Fehm for communications regarding connections of the paper’s subject matter to field theory and mathematical logic. The author was partially support by an AMS-Simons Travel Grant.


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Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA

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