Completions of Grothendieck groups

TL;DR

This paper introduces a method to define Euler characteristics for infinite projective resolutions within certain abelian categories by completing Grothendieck groups, enabling new applications in categorification.

Contribution

It develops a framework for completing Grothendieck groups to handle infinite resolutions and shows how functors induce continuous maps, advancing categorification techniques.

Findings

Defined Euler characteristic for infinite resolutions

Established continuous homomorphisms induced by functors

Applied framework to categorification examples

Abstract

For a certain class of abelian categories, we show how to make sense of the "Euler characteristic" of an infinite projective resolution (or, more generally, certain chain complexes that are only bounded above), by passing to a suitable completion of the Grothendieck group. We also show that right-exact functors (or their left-derived functors) induce continuous homomorphisms of these completed Grothendieck groups, and we discuss examples and applications coming from categorification.