The Abel Summation Method and Infinite Euler Characteristic

TL;DR

This paper introduces a new finiteness concept for unbounded chain complexes using Abel summation, defines their algebraic K-theory, and explores its relation to classical K-theory, revealing non-trivial structures.

Contribution

It develops a novel finiteness notion for unbounded complexes via Abel summation and establishes a connection between new and classical algebraic K-theory.

Findings

The algebraic K-theory of these complexes is non-trivial.

A natural map from classical to new K-theory contains a canonical infinite cyclic subgroup.

The paper demonstrates the relevance of Abel summation in algebraic K-theory.

Abstract

We develop a finiteness notion for unbounded chain complexes over a commutative noetherian integral domain $R$ employing the Abel summation method. The algebraic K-theory of such complexes is defined, and shown to be non-trivial. We also exhibit a natural map from the (usual) algebraic K-theory of $R$ into the new K-theory and show that its image contains a canonical infinite cyclic subgroup.