Hereditary cotorsion pairs and silting subcategories in extriangulated categories
type: Item
instance of: scholarly article
title: Hereditary cotorsion pairs and silting subcategories in extriangulated categories
zbMATH Open document ID: 1485.18020
author: Takahide Adachi
author: Mayu Tsukamoto
published in: Journal of Algebra
publication date: 2022-01-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/2108.07773
review text: The paper under review studies hereditary cotorsion pairs in extriangulated categories, and their relation with silting subcategories and co-t-structures. Let \((\mathcal{C},\mathbb{E},\mathfrak{s})\) be an extriangulated category, where \(\mathcal{C}\) is an additive \(R\)-linear category (\(R\) an associative ring with identity), \(\mathbb{E} \colon \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \longrightarrow \mathsf{Mod}(R)\) is an \(R\)-bilinear functor, and \(\mathfrak{s}\) is a realization of \(\mathbb{E}\). Two subcategories \(\mathcal{X}\) and \(\mathcal{Y}\) of \(\mathcal{C}\) form a (complete) cotorsion pair \((\mathcal{X,Y})\) if \(\mathcal{X}\) and \(\mathcal{Y}\) are closed under direct summands, \(\mathbb{E}(\mathcal{X,Y}) = 0\), and for every object \(C \in \mathcal{C}\) there exist two \(\mathfrak{s}\)-conflations \(Y \to X \to C \dashrightarrow\) and \(C \to Y' \to X' \dashrightarrow\) with \(X, X' \in \mathcal{X}\) and \(Y, Y' \in \mathcal{Y}\). If in addition \(\mathbb{E}^k(\mathcal{X,Y}) = 0\) for every \(k \geq 2\), then \((\mathcal{X,Y})\) is said to be hereditary. The collection of cotorsion pairs form a partially ordered set with the following relation: given \(x_1 = (\mathcal{X}_1,\mathcal{Y}_1)\) and \(x_2 = (\mathcal{X}_2,\mathcal{Y}_2)\), then \(x_1 \leq x_2\) if \(\mathcal{Y}_1 \subseteq \mathcal{Y}_2\). In this case, \([x_1,x_2]\) denotes the interval of cotorsion pairs \(x\) for which \(x_1 \leq x \leq x_2\). In their first main result (Theorem 3.6), the authors show the existence of an isomorphism between the interval \([x_1,x_2]\) and the poset \(\mathsf{cotors}(\mathcal{H}_{[x_1,x_2]})\) of cotorsion pairs in the coheart \(\mathcal{H}_{[x_1,x_2]} := \mathcal{X}_1 \cap \mathcal{Y}_2\). Namely, \(\Phi \colon [x_1,x_2] \to \mathsf{cotors}(\mathcal{H}_{[x_1,x_2]})\) is given by \(\Phi(\mathcal{X,Y}) := (\mathcal{X} \cap \mathcal{Y}_2,\mathcal{X}_1 \cap \mathcal{Y})\) with inverse \(\Psi \colon \mathsf{cotors}(\mathcal{H}_{[x_1,x_2]}) \t…
reviewed by: Marco A. Pérez
Mathematics Subject Classification ID: 18G80
zbMATH DE Number: 7459370
zbMATH Keywords: extriangulated categories
zbMATH Keywords: cotorsion pairs
zbMATH Keywords: silting subcategories
zbMATH Keywords: co-t-structures
MaRDI profile type: Publication
OpenAlex ID: W3194546214
arXiv ID: 2108.07773
cites work: Intervals of s-torsion pairs in extriangulated categories with negative first extensions
cites work: Silting mutation in triangulated categories
cites work: The homological theory of maximal Cohen-Macaulay approximations
cites work: Applications of contravariantly finite subcategories
cites work: Frobenius pairs in abelian categories. Frobenius pairs in abelian categories, correspondences with cotorsion pairs, exact model categories, and Auslander-Buchweitz contexts
cites work: Q4640173
cites work: Weight structures vs.t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)
cites work: How To Make Ext Vanish
cites work: \(n\)-exangulated categories (I): definitions and fundamental properties
cites work: Q3824559
cites work: Silting objects, simple-minded collections, \(t\)-structures and co-\(t\)-structures for finite-dimensional algebras.
cites work: Homological Theory of Representations
cites work: Hearts of twin cotorsion pairs on extriangulated categories
cites work: A new method to construct model structures from left Frobenius pairs in extriangulated categories
cites work: Auslander-Buchweitz context and co-\(t\)-structures
cites work: Q5226822
cites work: Compact corigid objects in triangulated categories and co-\(t\)-structures
cites work: Co-t-structures, cotilting and cotorsion pairs
cites work: The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences
cites work: Q3861607
cites work: One-sided Frobenius pairs in extriangulated categories
cites work: Tilting subcategories in extriangulated categories
DOI: 10.1016/J.JALGEBRA.2021.11.029
description: scientific article