Homological epimorphisms, compactly generated \(t\)-structures and Gorenstein-projective modules
type: Item
instance of: scholarly article
title: Homological epimorphisms, compactly generated \(t\)-structures and Gorenstein-projective modules
zbMATH Open document ID: 1394.18005
author: Nan Gao
author: Xiao Jing Xu
published in: Chinese Annals of Mathematics. Series B
publication date: 2018-02-14T00:00:00Z
review text: Given a set \(S\) of compact objects in a cocomplete triangulated category \(\mathcal{T}\), let \(\mathcal{D}'\) be the smallest full subcategory including \(S\) and closed under the suspension, extensions, coproducts and direct summands, and let \(\mathcal{D}''\) be the full subcategory of \(\mathcal{T}\) formed by the objects \(X''\) such that \(\text{Hom}_{\mathcal{T}}(X',X''[n]) = 0\), for all \(X' \in S\) and \(n \leq 0\). Then, \((\mathcal{D}',\mathcal{D}'')\) forms a \(t\)-structure on \(\mathcal{T}\), called \textit{compactly generated} (see [\textit{L. Alonso Tarrío} et al., Trans. Am. Math. Soc. 355, No. 6, 2523--2543 (2003; Zbl 1019.18007)], Theorem A.1). In the article [\textit{N. Broomhead} et al., J. Algebra 394, 51--78 (2013; Zbl 1322.18003)], the authors proved that, given a finite collection \(\{ (\mathcal{D}'_{i},\mathcal{D}''_{i}) \}\) of compactly generated \(t\)-structures on \(\mathcal{T}\), then the pair \((\mathcal{D}',\mathcal{D}'')\), where \(\mathcal{D}'\) is the smallest full subcategory of \(\mathcal{T}\) including all \(\mathcal{D}'_{i}\) and closed under extensions and direct summands, and \(\mathcal{D}''\) is the intersection of \(\{ \mathcal{D}''_{i} \}\), is a \(t\)-structure (see loc. cit. Theorem A). The latter is called the \textit{BPP-induction} of the previous collection of compactly generated \(t\)-structures in the article under review. On the other hand, it is a classical result that, given a recollement of triangulated categories of \(\mathcal{T}\) with respect to \(\mathcal{T}'\) and \(\mathcal{T}''\), and \(t\)-structures \((\mathcal{X}',\mathcal{X}'')\) on \(\mathcal{T}'\) and \((\mathcal{Y}',\mathcal{Y}'')\) on \(\mathcal{T}''\), then there is an induced \(t\)-structure on \((\mathcal{D}',\mathcal{D}'')\) on \(\mathcal{T}\) (see [\textit{A. A. Beilinson} et al., Astérisque 100, 172 p. (1982; Zbl 0536.14011)] , Theorem 1.4.10). The authors called this procedure the \textit{BBD-induction} of the \(t\)-structures \((\mathc…
reviewed by: Estanislao Herscovich
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16E35
zbMATH DE Number: 6838912
zbMATH Keywords: compactly generated \(t\)-structure
zbMATH Keywords: recollement
zbMATH Keywords: BBD-induction
zbMATH Keywords: BPP-induction
zbMATH Keywords: homological ring epimorphism
zbMATH Keywords: Gorenstein-projective module
MaRDI profile type: Publication
full work available at URL: s11401-018-1050-z
OpenAlex ID: W2781478145
cites work: Construction of 𝑡-structures and equivalences of derived categories
cites work: Recollements and tilting objects
cites work: Jordan-Hölder theorems for derived module categories of piecewise hereditary algebras.
cites work: Homological and homotopical aspects of torsion theories
cites work: Q4640173
cites work: Koszul duality
cites work: Averaging \(t\)-structures and extension closure of aisles
cites work: Good tilting modules and recollements of derived module categories
cites work: Stratifying endomorphism algebras
cites work: Relative singularity categories and Gorenstein-projective modules
cites work: Relative homological algebra
cites work: Gorenstein homological aspects of monomorphism categories via Morita rings
cites work: Perpendicular categories with applications to representations and sheaves
cites work: On Artin algebras arising from Morita contexts.
cites work: Recollements and Hochschild theory.
cites work: Reducing homological conjectures by \(n\)-recollements
cites work: Q2983723
cites work: Q3824559
cites work: Tilting complexes, perpendicular categories and recollements of derived module categories of rings
cites work: Hochschild cohomology and stratifying ideals.
cites work: Silting objects, simple-minded collections, \(t\)-structures and co-\(t\)-structures for finite-dimensional algebras.
cites work: Gluing Silting Objects
cites work: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability
cites work: Recollements and Gorenstein algebras
cites work: Gorenstein-projective modules over triangular matrix Artin algebras.
cites work: Unbounded ladders induced by Gorenstein algebras
DOI: 10.1007/S11401-018-1050-Z
description: scientific article; zbMATH DE number 6838912