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

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 ${\mathbb{Z}}_p$-extension. Our method of proof differs from these previous work. Their proof relies on an explicit description of certain Galois group via Kummer theory afforded by the context of a cyclotomic ${\mathbb{Z}}_p$-extension, whereas our approach is via considering the Iwasawa cohomology groups with coefficients in ${\mathbb{Z}}_p(i)$ for $i\ge 2$. 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 ${\mathbb{Z}}_p$-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 ${\mathbb{Z}}_p(i)$ for $i\ge 2$. Finally, we give examples of $p$-adic Lie extensions, where the second Iwasawa cohomology groups can have nontrivial $\mu$-invariants.

Some changes and additions

Cited by in corpus (4)

• On the codescent of étale wild kernels in $p$-adic Lie extensions
• Structure of fine Selmer groups over ${\mathbb{Z}}_p$-extensions
• Comparing direct limit and inverse limit of even $K$-groups in multiple ${\mathbb{Z}}_p$-extensions
• On the $p$-divisibility of even $K$-groups of the ring of integers of a cyclotomic field