# Asymptotic Grothendieck groups and c.b.l.f. positive dg-algebras

## TL;DR

This paper introduces asymptotic Grothendieck groups for certain categories, explores their properties, and computes these groups for specific positive dg-algebras with bounded, locally finite dimensions.

## Contribution

It defines asymptotic Grothendieck groups for abelian and triangulated categories and relates them to topological Grothendieck groups, providing new insights and computations for positive dg-algebras.

## Key findings

- Asymptotic Grothendieck groups are isomorphic under certain conditions.
- Connection established between asymptotic and topological Grothendieck groups.
- Computed asymptotic Grothendieck groups for specific positive dg-algebras.

## Abstract

We introduce the notion of asymptotic Grothendieck groups for abelian and triangulated categories that are both AB4 and AB4*. We study when the asymptotic Grothendieck group of the heart of a triangulated category with a t-structure is isomorphic to the asymptotic Grothendieck group of the triangulated category itself. We also explain a connexion with the notion of topological Grothendieck group from Achar-Stroppel. Finally, we compute the asymptotic Grothendieck group of some positive (dg-)algebras having a cone bounded, locally finite dimension.