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On the growth of even K-groups of rings of integers in p-adic Lie extensions

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Abstract

Let p be an odd prime number. In this paper, we study the growth of the Sylow p-subgroups of the even K-groups of rings of integers in a p-adic Lie extension. Our results generalize previous results of Coates and Ji—Qin, where they considered the situation of a cyclotomic ℤp-extension. Our method of proof differs from these previous works. Their proof relies on an explicit description of certain Galois group via Kummer theory afforded by the context of a cyclotomic ℤp-extension, whereas our approach is via considering the Iwasawa cohomology groups with coefficients in ℤp (i) for i ≥ 2. We should mention that this latter approach is possible thanks to the Quillen—Lichtenbaum Conjecture which is now known to be valid by the works of Rost—Voevodsky. We also note that the approach allows us to work with more general p-adic Lie extensions that do not necessarily contain the cyclotomic ℤp-extension, where the Kummer theoretical approach does not apply. Along the way, we study the torsionness of the second Iwasawa cohomology groups with coefficients in ℤp (i) for i ≥ 2. Finally, we give examples of p-adic Lie extensions, where the second Iwasawa cohomology groups can have nontrivial μ-invariants.

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Acknowledgement

The author would like to thank John Coates and Antonio Lei for their interest and comments. He also liked to thank the anonymous referee for the many valuable comments and for pointing out some inaccuracies in a previous draft of the paper. This research is supported by the National Natural Science Foundation of China under Grant No. 11550110172 and Grant No. 11771164.

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  1. School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, P.R. China

    Meng Fai Lim

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Lim, M.F. On the growth of even K-groups of rings of integers in p-adic Lie extensions. Isr. J. Math. 249, 735–767 (2022). https://doi.org/10.1007/s11856-022-2324-4

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