Extending coherent and quasi-coherent sheaves on generically closed spaces
type: Item
instance of: scholarly article
title: Extending coherent and quasi-coherent sheaves on generically closed spaces
zbMATH Open document ID: 0545.14011
DOI: 10.1016/0021-8693(84)90241-2
author: Fred Van Oystaeyen
author: Alain Verschoren
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: Both the Picard group of a quasi-affine scheme and the class group of a normal domain may be described in a module theoretic setting as special cases of the so-called relative Picard group of a ring R associated to an idempotent kernel functor on R-mod. This paper aims to show that similar methods apply in a much more general context. Let us give some examples. It is well known that reflexive Modules (i.e., sheaves of modules) over an integral normal scheme X are locally free sheaves of rank one over \(X^{(1)}\), the subspace of points of codimension 1 of X. Moreover the class group of X is just the Picard group of \(X^{(1)}\). Another example is obtained by considering the local scheme at \(x\in X\) for some scheme X; i.e. consider the scheme Spe\(c({\mathcal O}_{X,x})\) canonically embedded in X. As a set this local scheme at x consists of all \(y\in X\) such that x is in the closure of \(\{\) \(y\}\) in X. - The general problem is the following: Given some subspace \(Y\subset X\), for example, an open subspace or one of the above mentioned, classify the extensions of a quasi-coherent sheaf on Y with structure sheaf \({\mathcal O}_ x| Y\) to a quasi-coherent sheaf on X. This problem arises in the description of relative invariants with respect to Y. If Y is open in X and \({\mathcal M}\) is quasi-coherent on Y then \(i_*{\mathcal M}\) is quasi-coherent on X, where i:\(Y\to X\) is the canonical inclusion. If X is Noetherian (we will assume this throughout) then a result of A. Grothendieck states that any coherent sheaf on Y may be extended to a coherent sheaf on X (note that \(i_*{\mathcal M}\) is not necessarily coherent itself). However, \(X^{(1)}\) is not even a subscheme of X, so in the situations we want to deal with Grothendieck's result cannot be applied. The subspaces Y we consider will be geometrically stable (generically closed), i.e. \(y\in Y\) and \(x\in X\), with \(y\in \{\bar x\}\), the closure of \(\{\) \(x\}\) in X, imply \(x\in Y\). The advantage o…
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14C22
Mathematics Subject Classification ID: 13D15
zbMATH DE Number: 3867486
zbMATH Keywords: generically closed subspace
zbMATH Keywords: geometrically stable subspace
zbMATH Keywords: Picard group of a quasi-affine scheme
zbMATH Keywords: class group of a normal domain
zbMATH Keywords: extensions of a quasi-coherent sheaf
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90241-2
OpenAlex ID: W1991001678
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description: scientific article; zbMATH DE number 3867486