论文标题
关于甚至$ k $ - $ k $ groups of $ p $ p $ addic lie扩展的增长
On the growth of even $K$-groups of rings of integers in $p$-adic Lie extensions
论文作者
论文摘要
令$ p $为奇数。在本文中，我们研究了Sylow $ p $ -subgroups在$ p $ adiC lie扩展中的什至$ k $ - 组的增长。我们的结果概括了Coates和Ji-Qin的先前结果，在那里他们考虑了Cyclotomic $ \ Mathbb {Z} _P $ -Extension的情况。我们的证明方法与以前的工作不同。他们的证明依赖于通过Kummer理论对某些Galois组的明确描述，该理论是由循环组合$ \ Mathbb {z} _p $ - extension提供的，而我们的方法是通过考虑iWasawa cohomology群体，其系数在$ \ Mathbb {z} _p（z} _p（i）$ geq for $ i \ y i \ i \ i \ y i \ i \ y y \ i \ y y \ y \ i \ i \ i \ i \ i \ y \ i \ y \ i \ y \ i \ y \ i。我们应该提到，由于Quillen-Lichtebaum的猜想，这是后一种方法，现在众所周知，Rost-Voevodsky的作品有效。我们还注意到，该方法使我们能够使用更通用的$ p $ -Adic Lie扩展名，这些扩展不一定包含Cyclotomic $ \ Mathbb {z} _p $ - extension，其中Kummer理论方法不适用于。在此过程中，我们研究了第二个Iwasawa共同学组的扭转，其系数为$ \ Mathbb {z} _p（i）$ for $ i \ geq 2 $。最后，我们举例说明了$ p $ adadic lie扩展名，第二个Iwasawa共同学组可以具有非平凡的$μ$ invariants。
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 $i\geq 2$. 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 $i\geq 2$. Finally, we give examples of $p$-adic Lie extensions, where the second Iwasawa cohomology groups can have nontrivial $μ$-invariants.
