Higher arithmetic $K$-theory

Adams Operations on Higher Arithmetic K-theory

We construct Adams operations on the rational higher arithmetic K-groups of a proper arithmetic variety. The definition applies to the higher arithmetic K-groups given by Takeda as well as to the groups suggested by Deligne and Soulé, by means of the homotopy groups of the homotopy fiber of the regulator map. They are compatible with the Adams operations on algebraic K-theory. The definition re...

On Higher Arithmetic Intersection Theory

Higher K - Theory Speaker :

Notes from the second meeting of the algebraic K-theory seminar at UMich, Winter 2015. Note taker was Takumi Murayama. Motivation One reason we would like to define higher K-groups is that with K0, we get certain “half-exact” sequences that seem to be analogues of the Mayer-Vietoris or the localization long exact sequences, that we would like to actually turn into long exact sequences. Historic...

Arithmetic Teichmuller Theory

By Grothedieck's Anabelian conjectures, Galois representations landing in outer automorphism group of the algebraic fundamental group which are associated to hyperbolic smooth curves defined over number fields encode all arithmetic information of these curves. The goal of this paper is to develope and arithmetic teichmuller theory, by which we mean, introducing arithmetic objects summarizing th...

Higher K-theory of Koszul cubes

In this paper, for noetherian commutative ring with a unit A such that every finitely generated A-module is free, and for a certain A-regular sequence f1, · · · , fp, we define Kos(A; f1, · · · , fp) the category of Koszul cubes associated to f1, · · · , fp which has a natural Waldhausen categorical structure. We also give a new algorithm of resolution process of modules by Koszul cubes. By the...