# Metric completions of triangulated categories from hereditary rings

## TL;DR

This paper investigates the metric completions of triangulated categories associated with hereditary rings, providing explicit descriptions for certain classes and linking these completions to universal localisations.

## Contribution

It develops a lattice theory of metrics on triangulated categories and describes all completions for specific classes of hereditary rings, connecting them to ring localisations.

## Key findings

- Explicit descriptions of metric completions for hereditary commutative noetherian rings.
- Explicit descriptions of metric completions for hereditary algebras of tame representation type.
- Established a connection between metric completions and universal localisations.

## Abstract

The focus of this article is on metric completions of triangulated categories arising in the representation theory of hereditary finite dimensional algebras and commutative rings. We explicitly describe all completions of bounded derived categories with respect to additive good metrics for two classes of rings - hereditary commutative noetherian rings and hereditary algebras of tame representation type over an algebraically closed field. To that end, we develop and study the lattice theory of metrics on triangulated categories. Moreover, we establish a link between metric completions of bounded derived categories of a ring and the ring's universal localisations.