Descent and ascent of local properties along homomorphisms of finite flat dimension
type: Item
instance of: scholarly article
title: Descent and ascent of local properties along homomorphisms of finite flat dimension
zbMATH Open document ID: 0603.13007
DOI: 10.1016/0022-4049(85)90007-6
author: Luchezar L. Avramov
author: Hans-Bjørn Foxby
author: Stephen Halperin
published in: Journal of Pure and Applied Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let \(f: R\to S\) be a local homomorphsm of Noetherian local rings. It is well known that if f is flat then the property of being (1) regular, (2) complete intersection, (3) Gorenstein, (4) Cohen-Macaulay, respectively, descends from S to R. In fact, properties (1), (2) and (4) are known to descend even under the weaker condition that f makes S an R-module of finite flat dimension. One of the results proved in this paper is that (3) also descends under this weaker condition. Moreover, a quantitative expression is provided to these results by proving that if f makes S an R-module of finite flat dimension then appropriate inequalities hold between certain numerical functions of S measuring its deviation from one of the above properties and the corresponding functions of R. The authors also consider the ascent of the above properties. Assuming that f makes S an R-module of finite flat dimension, conditions are found on the homotopy fibre of f which ensure, respectively, the ascent of properties (1), (2) and (3) from R to S.
Mathematics Subject Classification ID: 13B99
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 14B25
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 13H05
zbMATH DE Number: 3973007
zbMATH Keywords: descent
zbMATH Keywords: local homomorphsm of Noetherian local rings
zbMATH Keywords: ascent
Wikidata QID: Q102254643
reviewed by: Balwant Singh
MaRDI profile type: Publication
full work available at URL: 0022-4049(85)90007-6
OpenAlex ID: W1980138772
cites work: Localisation de la lissite formelle
cites work: Le caractère additif des déviations des anneaux locaux
cites work: Homology of local flat extensions and complete intersection defects
cites work: Q4748239
cites work: Q3344077
cites work: Homology algebra of the Koszul complex of a local Gorenstein ring
cites work: Algebra Structures for Finite Free Resolutions, and Some Structure Theorems for Ideals of Codimension 3
cites work: On the $\mu^i$ in a minimal injective resolution II.
cites work: Bounded complexes of flat modules
cites work: Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. (Séconde partie)
cites work: Q5610428
cites work: Rational Fibrations, Minimal Models, and Fibrings of Homogeneous Spaces
cites work: Q4055756
cites work: Inequalities related to certain couples of local rings
Recommended article: \(F\)-finiteness of homomorphisms and its descent
Recommended article: Flat local morphisms of rings with prescribed depth and dimension
Recommended article: Russian text ignore
Recommended article: Descendents on local curves: Stationary theory
Recommended article: Q3385990
Recommended article: Descent of the complete intersection property by homomorphisms of finite virtual projective dimension
Recommended article: Topological properties of localizations, flat overrings and sublocalizations
Recommended article: Q4488136
Recommended article: Q3562171
Recommended article: Q4615810
description: scientific article; zbMATH DE number 3973007