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Metric completions of triangulated categories from finite dimensional algebras

Cyril Matoušek

TL;DR

The paper investigates metric completions of triangulated categories, focusing on bounded derived categories $\mathbf{D}^b(\mathrm{mod}\,A)$ for hereditary finite-dimensional algebras $A$ of finite representation type, and provides an explicit completion description in terms of the intersection of the metric balls $\mathcal{B}=\bigcap_{n}B_n$ via $\mathcal{B}^\perp$.A key innovation is the use of image and preimage metrics under triangulated functors to transfer completions across categories and to establish triangulated equivalences between completions, even beyond the good-metric setting.The paper introduces the notion of an improvement of a metric, proving that every metric has a naturally associated good metric and that the original completion embeds into the improvement-based completion, clarifying the relation between generic and good metrics.In the presence of Serre functors (e.g. when the source algebra has finite global dimension), the authors extend these transport-and-equivalence ideas to connect completions along adjoint pairs and provide concrete examples for Dynkin-type derived categories, including a demonstration with the $A_2$ Dynkin quiver and related functors.Overall, the work advances the computational toolkit for metric completions in representation theory, linking thick subcategories with completions and enabling explicit calculations in Dynkin- and representation-finite settings.

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.

Metric completions of triangulated categories from finite dimensional algebras

TL;DR

The paper investigates metric completions of triangulated categories, focusing on bounded derived categories $\mathbf{D}^b(\mathrm{mod}\,A)$ for hereditary finite-dimensional algebras $A$ of finite representation type, and provides an explicit completion description in terms of the intersection of the metric balls $\mathcal{B}=\bigcap_{n}B_n$ via $\mathcal{B}^\perp$.A key innovation is the use of image and preimage metrics under triangulated functors to transfer completions across categories and to establish triangulated equivalences between completions, even beyond the good-metric setting.The paper introduces the notion of an improvement of a metric, proving that every metric has a naturally associated good metric and that the original completion embeds into the improvement-based completion, clarifying the relation between generic and good metrics.In the presence of Serre functors (e.g. when the source algebra has finite global dimension), the authors extend these transport-and-equivalence ideas to connect completions along adjoint pairs and provide concrete examples for Dynkin-type derived categories, including a demonstration with the $A_2$ Dynkin quiver and related functors.Overall, the work advances the computational toolkit for metric completions in representation theory, linking thick subcategories with completions and enabling explicit calculations in Dynkin- and representation-finite settings.

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.
Paper Structure (18 sections, 45 theorems, 71 equations)

This paper contains 18 sections, 45 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and conventions
  4. General category theory
  5. Additive categories
  6. Triangulated categories
  7. Homotopy categories and derived categories
  8. Rings and modules
  9. Metrics on triangulated categories
  10. Completions of triangulated categories
  11. Good extensions
  12. Completions of certain derived categories
  13. Functors between completions
  14. Induced functors on completions
  15. Preimages and images of metrics
  16. ...and 3 more sections

Key Result

theorem 1

Let $A$ be a hereditary finite dimensional algebra (over a field) of finite representation type. Let $\{B_n\}_{n\in\N}$ be a good metric on the category $\derived^b(\modf\dashmodule A)$. Denote $\mathcal{B}:=\bigcap_{n=1}^\infty B_n$. The completion of $\derived^b(\modf\dashmodule A)$ with respect t

Theorems & Definitions (104)

  • theorem 1: Theorem \ref{['DynkinCompletions']}
  • theorem 2: SunZhang21, Theorem \ref{['SunZhangTheorem']}
  • theorem 3: Theorem \ref{['CalculatingCompletionsElsewhere']}
  • definition 1
  • definition 2: Neeman18
  • remark 1
  • definition 3: Neeman18
  • remark 2: Neeman18
  • remark 3
  • definition 4: Neeman18
  • ...and 94 more