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Igusa-Todorov properties of recollements of abelian categories

Peiru Yang, Yajun Ma, Yu-Zhe Liu

TL;DR

This work studies how Igusa-Todorov properties, encoded by $(m,n)$-Igusa-Todorov structures and the $IT.dist$ distance, behave under recollements of abelian categories. The authors develop a framework to transfer $IT$-type finiteness across a recollement, deriving bounds for the middle category’s $IT.dist$ in terms of the outer categories under exactness hypotheses, and then apply these results to Artin algebras, Morita context rings, and idempotent-induced recollements. Key contributions include general theorems bounding $IT.dist(\mathscr{B})$ by $IT.dist(\mathscr{A})$ and $IT.dist(\mathscr{C})$, plus concrete corollaries for algebras built from idempotents and Morita contexts, and examples involving triangular matrix rings. The findings offer a practical pathway to deduce homological finiteness properties of a complex algebra from its components, with implications for representation theory and the study of singularity categories.

Abstract

In this paper, we investigate the behavior of Igusa-Todorov properties under recollements of abelian categories. In particular, we study how the Igusa-Todorov distances of the categories involved in a recollement are related. Applications are given to Artin algebras, especially to Morita context rings.

Igusa-Todorov properties of recollements of abelian categories

TL;DR

This work studies how Igusa-Todorov properties, encoded by -Igusa-Todorov structures and the distance, behave under recollements of abelian categories. The authors develop a framework to transfer -type finiteness across a recollement, deriving bounds for the middle category’s in terms of the outer categories under exactness hypotheses, and then apply these results to Artin algebras, Morita context rings, and idempotent-induced recollements. Key contributions include general theorems bounding by and , plus concrete corollaries for algebras built from idempotents and Morita contexts, and examples involving triangular matrix rings. The findings offer a practical pathway to deduce homological finiteness properties of a complex algebra from its components, with implications for representation theory and the study of singularity categories.

Abstract

In this paper, we investigate the behavior of Igusa-Todorov properties under recollements of abelian categories. In particular, we study how the Igusa-Todorov distances of the categories involved in a recollement are related. Applications are given to Artin algebras, especially to Morita context rings.
Paper Structure (6 sections, 22 theorems, 100 equations)

This paper contains 6 sections, 22 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions
  4. Recollements of abelian categories
  5. Igusa-Todorov properties
  6. Applications and examples

Key Result

Theorem 1.1

Let $(\mathscr{A},\mathscr{B},\mathscr{C})$ be a recollement of abelian categories with $\mathsf{Proj}\mathscr{B}=\mathop{\rm add}\nolimits(P)$ for some projective object $P$. Assume that $\mathscr{A}$ is an $(m,r)$-Igusa-Todorov category and $\mathscr{C}$ is an $(n,s)$-Igusa-Todorov category. Then

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Example 2.3
  • Example 2.4
  • Definition 3.1
  • Remark 3.2
  • ...and 43 more