$K_{i}^{loc}(\mathbb{C})$, $i = 0, 1$

TL;DR

This paper computes the local algebraic K-theory groups for the complex numbers with trivial filtration, contributing to a broader program to reformulate the index theorem in a local algebraic context.

Contribution

It provides explicit calculations of local algebraic K-theory for complex numbers, advancing the understanding of local K-theory in the simplest case.

Findings

Computed local algebraic K-theory groups for  with trivial filtration

Established the case as a basic example of local K-theory over a point

Contributed to the program of re-stating the index theorem using local algebraic K-theory

Abstract

In this article we compute the {\em local algebraic $K$-theory}, $ i = 0, 1$, of the algebra of complex numbers $\mathbb{C}$ endowed with the trivial filtration, i.e. $\mathbb{C}_{\mu}= \mathbb{C}$, for any $\mu \in \mathbb{N}$; {\em local algebras} and {\em local} algebraic $K^{loc}_{i}$-theory were introduced in \cite{Teleman_arXiv_IV}. \par Theorem 3 states the result. \par This case corresponds in the simplest case to the Alexander-Spanier {\em local $K$-theory} over the point. \par This article is part of a comprehensive program aimed at re-stating the index theorem, see \cite{Teleman_arXiv_III}. Other articles in this series are \cite{Teleman_arXiv_IV}, \cite{Teleman_arXiv_I}, \cite{Teleman_arXiv_II}.