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Wakamatsu tilting subcategories and weak support tau-tilting subcategories in recollements

Yongduo Wang, Hongyang Luo, Yu-Zhe Liu, Jian He, Dejun Wu

TL;DR

The paper studies how tilting-like structures behave across recollements of abelian categories $(\mathcal{A}, \mathcal{B}, \mathcal{C})$. It proves that Wakamatsu tilting subcategories in the outer categories glue to a Wakamatsu tilting subcategory in the middle category, and that a converse holds under natural exactness assumptions; the same is shown for weak support $\tau$-tilting subcategories. As an application, the authors relate $\tau$-cotorsion torsion triples across recollements via the established bijection with support $\tau$-tilting subcategories. The results are illustrated with explicit module-category examples that realize the gluing constructions in concrete algebras, providing practical insights into transferring tilting-like structures through recollements.

Abstract

In this article, we prove that if (A, B, C) is a recollement of abelian categories, then Wakamatsu tilting (resp. weak support tau-tilting) subcategories in A and C can induce Wakamatsu tilting (resp. weak support tau-tilting) subcategories in B, and the converses hold under natural assumptions. As an application, we mainly consider the relationship of tau-cotorsion torsion triples in (A, B, C).

Wakamatsu tilting subcategories and weak support tau-tilting subcategories in recollements

TL;DR

The paper studies how tilting-like structures behave across recollements of abelian categories . It proves that Wakamatsu tilting subcategories in the outer categories glue to a Wakamatsu tilting subcategory in the middle category, and that a converse holds under natural exactness assumptions; the same is shown for weak support -tilting subcategories. As an application, the authors relate -cotorsion torsion triples across recollements via the established bijection with support -tilting subcategories. The results are illustrated with explicit module-category examples that realize the gluing constructions in concrete algebras, providing practical insights into transferring tilting-like structures through recollements.

Abstract

In this article, we prove that if (A, B, C) is a recollement of abelian categories, then Wakamatsu tilting (resp. weak support tau-tilting) subcategories in A and C can induce Wakamatsu tilting (resp. weak support tau-tilting) subcategories in B, and the converses hold under natural assumptions. As an application, we mainly consider the relationship of tau-cotorsion torsion triples in (A, B, C).
Paper Structure (6 sections, 18 theorems, 85 equations, 2 figures)

This paper contains 6 sections, 18 theorems, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gluing Wakamatsu tilting subcategories
  4. Gluing weak support $\tau$-tilting subcategories
  5. Application
  6. Examples

Key Result

Lemma 2.2

Let ($\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$) be a recollement of abelian categories as (recolle). $(1)$ All the natural transformations are natural isomorphisms. $(2)$$i^{\ast}j_!=0$ and $i^{!}j_\ast=0$. $(3)$ If $i^{\ast}$ (resp. $i^{!}$) is exact, then ${i^!}{j_!} = 0$ (resp. ${i^*}{j_*} = 0$). $(4)$ If $i^{\ast}$ (resp. $i^{!}$) is exact, then $j_{!}$ (resp. $j_{\ast}$) is exact. $(5)$ Fo

Figures (2)

  • Figure 1: The marked surface $\mathbf{S}$ of $A$
  • Figure 3: A support $\tau$-tilting module

Theorems & Definitions (38)

  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • Remark 2.5
  • Lemma 2.6
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • ...and 28 more