Metric completions of triangulated categories from hereditary rings
Cyril Matoušek
TL;DR
This paper develops a systematic framework for metric completions of triangulated categories arising from hereditary rings, focusing on the bounded derived category $D^b( ext{mod-}R)$. It introduces and leverages a lattice-theoretic view of additive good metrics, coupling completion data with universal localisations and smashing/recolllement structures to obtain explicit descriptions. For commutative noetherian hereditary rings (via Dedekind components) and for hereditary tame algebras (Euclidean quivers), the authors classify completions as either derived categories of universal localisations or as thick subcategories defined by perpendicularity to a fixed thick subcategory, with the latter capturing new objects that do not lie in the original category. A central theme is the translation of metric convergence properties into algebraic localisations, yielding computable descriptions of completions and revealing when completions remain inside the original category versus when new objects appear. The results provide concrete tools for constructing and understanding completions in representation theory and related homological contexts, and they illuminate how universal localisation and recollement phenomena govern the compactly supported nature of completed objects.
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.