Metric completions of triangulated categories from finite dimensional algebras

TL;DR

This paper explores metric completions of triangulated categories derived from finite dimensional algebras, providing explicit descriptions and methods to compare different completions in a representation-theoretic setting.

Contribution

It introduces a framework for understanding metric completions of derived categories, including new constructions like the improvement metric and their equivalences.

Findings

Explicit description of completions for hereditary finite dimensional algebras

Definition of image and preimage metrics under triangulated functors

Construction of a new good metric called the improvement

Abstract

In this paper, we study metric completions of triangulated categories in a representation-theoretic context. We provide a concrete description of completions of bounded derived categories of hereditary finite dimensional algebras of finite representation type. In order to investigate completions of bounded derived categories of algebras of finite global dimension, we define image and preimage metrics under a triangulated functor and use them to induce a triangulated equivalence between two completions. Furthermore, for a given metric on a triangulated category we construct a new, closely related good metric called the improvement and compare the respective completions.