The homological theory of quasi-resolving subcategories
type: Item
instance of: scholarly article
title: The homological theory of quasi-resolving subcategories
zbMATH Open document ID: 1386.18048
author: Xiaosheng Zhu
published in: Journal of Algebra
publication date: 2014-09-02T00:00:00Z
review text: In [Mem. Am. Math. Soc. 94, 146 p. (1969; Zbl 0204.36402)], \textit{M. Auslander} and \textit{M. Bridger} introduced the notion of resolving subcategories, capturing the behaviour of several natural subcategories of module categories in this notion. The starting point of the article under review is the observation that the nice homological behaviour of resolving subcategories extends also to setups where the definition does not apply. Xiaosheng Zhu then proceeds to remedy this situation, generalising the definition of resolving subcategories by introducing ``quasi-resolving subcategories''. (Essentially the generation-assumption is weakened, and it is only required that the projectives inside the subcategory generate the subcategory.) It is shown that many homological properties of resolving subcategories extend naturally to this more general notion.
reviewed by: Steffen Oppermann
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 18G20
Mathematics Subject Classification ID: 18G25
Mathematics Subject Classification ID: 16E05
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16E30
zbMATH DE Number: 6338797
zbMATH Keywords: quasi-resolving subcategories
zbMATH Keywords: quasi-resolving resolution dimensions
zbMATH Keywords: Syzygy
zbMATH Keywords: Wakamatsu tilting modules
zbMATH Keywords: relative cohomology
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2014.05.018
OpenAlex ID: W2071628483
cites work: Homological invariants associated to semi-dualizing bimodules.
cites work: Stable module theory
cites work: Applications of contravariantly finite subcategories
cites work: ABSOLUTE, RELATIVE, AND TATE COHOMOLOGY OF MODULES OF FINITE GORENSTEIN DIMENSION
cites work: On Gorenstein projective, injective and flat dimensions -- a functorial description with applications
cites work: Relative homological algebra. Vol. 1
cites work: Gorenstein injective and projective modules
cites work: Gorenstein Modules and Related Modules.
cites work: Dualities on generalized Koszul algebras
cites work: Syzygy theorems via comparison of order ideals on a hypersurface
cites work: Foxby equivalence over associative rings.
cites work: Gorenstein homological dimensions.
cites work: Semi-dualizing modules and related Gorenstein homological dimensions
cites work: Q4674273
cites work: G-dimension, complete intersection dimension, complexity, and relative homological algebra
cites work: Wakamatsu tilting modules.
cites work: An introduction to homological algebra
cites work: AB-contexts and stability for Gorenstein flat modules with respect to semidualizing modules
cites work: Stability of Gorenstein categories
cites work: On modules with trivial self-extensions
cites work: Tilting modules and Auslander's Gorenstein property.
cites work: Homological aspects of semidualizing modules
cites work: Classifying thick subcategories of the stable category of Cohen-Macaulay modules
cites work: Gorenstein projective dimension with respect to a semidualizing module
cites work: Flat covers of modules
cites work: Resolving resolution dimensions
DOI: 10.1016/J.JALGEBRA.2014.05.018
description: scientific article; zbMATH DE number 6338797