Comparing direct limit and inverse limit of even $K$-groups in non-commutative $p$-adic Lie extensions

TL;DR

The paper extends a duality between direct and inverse limits of higher even K-groups from commutative $p$-adic Lie extensions to certain non-commutative cases by connecting $K$-groups to Galois cohomology via Soulé– Rost–Voevodsky p-adic Chern classes and employing non-commutative Iwasawa theory. It develops the framework of Iwasawa cohomology for non-abelian extensions, leverages Lazard’s theory of Iwasawa algebras, and uses Tate descent and Jannsen-type exact sequences to bound Ext^1 groups, enabling a duality: $(\varinjlim_L K_{2i-2}(O_{L,S})[p^\infty])^\vee \cong Ext^1_{Z_p[[G]]}(\varprojlim_L K_{2i-2}(O_{L,S})[p^\infty], Z_p[[G]])$ under two concrete hypotheses. The results are obtained first for totally real base fields (case a) and then for the multi-false Tate extension (case b), with (b) crucially hinging on a Kato-type filtration and cohomological descent arguments. Collectively, the work broadens duality phenomena for higher K-theory into non-commutative Iwasawa theory, linking $K$-theory and Galois cohomology in new non-abelian settings.

Abstract

In a previous paper of the author, we establish a duality for the direct limit and the inverse limit of higher even $K$-groups over a $\mathbb{Z}_p^d$-extension. In this paper, we shall establish such a duality over certain non-commutative $p$-adic Lie extensions.

Theorems & Definitions (19)

• Theorem : LimKlimit
• Theorem 1.1
• Definition 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Theorem 3.1
• Proposition 3.2
• Remark 3.3