Projective modules over rings of global sections of flasque sheaves
type: Item
instance of: scholarly article
title: Projective modules over rings of global sections of flasque sheaves
zbMATH Open document ID: 0708.13002
DOI: 10.1016/0022-4049(90)90118-2
author: Sergey Yuzvinsky
published in: Journal of Pure and Applied Algebra
publication date: 1990-01-01T00:00:00Z
review text: Let (X,\({\mathcal O})\) be a quasicompact ringed space for which the sheaf \({\mathcal O}\) is flasque and locally Serre. (A ring is Serre if all projective modules of finite type are free.) The present paper proves (corollary 1.3) that the global sections functor \(\Gamma\) gives an equivalence of categories between the category of locally free (left) \({\mathcal O}\)-modules of finite rank and the category of projective finitely generated (left) \(\Gamma\) (\({\mathcal O})\)-modules. A sufficient condition is given (corollary 2.2) for \(\Gamma\) (\({\mathcal O})\) to be Serre. From this the author concludes (theorem 3.1) that the face ring of a finite simplicial complex over an arbitrary field is Serre, thus reproving a result of \textit{T. Vorst} [Math. Z. 184, 425-433 (1983; Zbl 0504.13008)].
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 4163036
zbMATH Keywords: flasque sheaves
zbMATH Keywords: flabby sheaf
zbMATH Keywords: freeness of projective module
zbMATH Keywords: Serre ring
zbMATH Keywords: global sections
zbMATH Keywords: face ring
reviewed by: Leslie G. Roberts
MaRDI profile type: Publication
full work available at URL: 0022-4049(90)90118-2
OpenAlex ID: W2020450363
cites work: Q5632701
cites work: Q3247612
cites work: Q4143433
cites work: Introduction to Algebraic K-Theory. (AM-72)
cites work: Combinatorics and commutative algebra
cites work: The Serre problem for discrete Hodge algebras
cites work: Cohen-Macaulay rings of sections
cites work: Flasque sheaves on posets and Cohen-Macaulay unions of regular varieties
description: scientific article; zbMATH DE number 4163036