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

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Abstract

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.

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Notes

  1. Another definition of unramified cohomology that is frequently encountered is

    $$\begin{aligned} H^i_{\mathrm {nr}} (F(X), \mu _m^{\otimes j}) = H^i (F(X), \mu _m^{\otimes j})_{V_1}, \end{aligned}$$

    where \(V_1\) is the set of all discrete valuations v of F(X) such that F is contained in the valuation ring \(\mathcal {O}_v\). Clearly, there is an inclusion \(H^i_{\mathrm {nr}} (F(X), \mu _m^{\otimes j}) \subset H^i_{\mathrm {ur}} (F(X), \mu _m^{\otimes j})\), which is in fact an equality if X is proper (see [9, Theorem 4.1.1]). Note that by construction, \(H^i_{\mathrm {nr}} (F(X), \mu _m^{\otimes j})\) is a birational invariant of X.

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Acknowledgements

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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  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Igor A. Rapinchuk

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Rapinchuk, I.A. A generalization of Serre’s condition \(\mathrm {(F)}\) with applications to the finiteness of unramified cohomology. Math. Z. 291, 199–213 (2019). https://doi.org/10.1007/s00209-018-2079-0

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