Comparing direct limit and inverse limit of even $K$-groups in multiple $\mathbb{Z}_p$-extensions

TL;DR

This paper extends Iwasawa's duality between direct and inverse limits from class groups to higher even $K$-groups in multiple $Z_p$-extensions, revealing new duality relations in algebraic $K$-theory.

Contribution

It establishes an analogous duality for higher even $K$-groups in $Z_p^d$-extensions, generalizing previous results from class groups.

Findings

Duality between direct and inverse limits of higher even $K$-groups

Extension of Iwasawa duality to multiple $Z_p$-extensions

New relations in algebraic $K$-theory for $Z_p^d$-extensions

Abstract

Iwasawa first established a duality relating the direct limit and the inverse limit of class groups in a $\mathbb{Z}_p$-extension, and this result has recently been extended to multiple $\mathbb{Z}_p$-extensions by many authors. In this paper, we establish an analogous duality for the direct limit and the inverse limit of higher even $K$-groups in a $\mathbb{Z}_p^d$-extension.