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$n$-cotorsion pairs and recollements of extriangulated categories

Jian He, Jing He

Abstract

In this article, we prove that if $(\mathcal A ,\mathcal B,\mathcal C)$ is a recollement of extriangulated categories, then $n$-cotorsion pairs in $\mathcal A$ and $\mathcal C$ can induce $n$-cotorsion pairs in $\mathcal B$. Conversely, this holds true under natural assumptions. Besides, we give mild conditions on a pseudo cluster tilting subcategory on the middle category of a recollement of extriangulated categories, for the corresponding additive quotients to form a recollement of semi-abelian categories.

$n$-cotorsion pairs and recollements of extriangulated categories

Abstract

In this article, we prove that if is a recollement of extriangulated categories, then -cotorsion pairs in and can induce -cotorsion pairs in . Conversely, this holds true under natural assumptions. Besides, we give mild conditions on a pseudo cluster tilting subcategory on the middle category of a recollement of extriangulated categories, for the corresponding additive quotients to form a recollement of semi-abelian categories.
Paper Structure (4 sections, 15 theorems, 35 equations)

This paper contains 4 sections, 15 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Glued $n$-cotorsion pairs
  4. From recollement of extriangulated categories to recollement of semi-abelian categories

Key Result

Lemma 2.2

Let $\mathcal{C}$ be an extriangulated category with enough projectives and enough injectives. Assume that \xymatrix{A\ar[r]^{f}& B\ar[r]^{g}&C\ar@{-->}[r]^{\delta}&}is an $\mathbb{E}$-triangle in $\mathcal{C}$. Then for any object $X\in\mathcal{C}$ and $k\geq1$, we have the following exact sequence

Theorems & Definitions (24)

  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • Lemma 3.4
  • Theorem 3.5
  • ...and 14 more