Global sections of diffspec
type: Item
instance of: scholarly article
title: Global sections of diffspec
zbMATH Open document ID: 1029.13014
DOI: 10.1016/S0022-4049(01)00118-9
author: Jerald J. Kovacic
published in: Journal of Pure and Applied Algebra
publication date: 2002-08-22T00:00:00Z
review text: Let \(\mathcal R\) be a differential ring, i.e., \(\mathcal R\) is a commutative ring together with a finite set \(\Delta\) of pairwise commuting differentials. Let \(X=\text{diffspec}({\mathcal R})\) be the set of all prime differential ideals of \(\mathcal R\); in the usual way one defines a sheaf \({\mathcal O}_X\) of \(\Delta\)-rings and a canonical homomorphism \(\iota_{\mathcal R} : { \mathcal R }\to \Gamma (X, {\mathcal O}_X) = : \widehat{\mathcal R}\) of the ring \({\mathcal R}\) in the ring of global sections of the sheaf \({\mathcal O}_X\), and, for every \(\Delta\)-\(\mathcal R\)-module \(\mathcal M\), a sheaf \(\widetilde{\mathcal M}\) of \(\Delta\)-\({\mathcal O}_X\)-modules and a canonical homomorphism of \(\Delta \)-\({\mathcal R}\)-modules \(\iota_{\mathcal M} : {\mathcal M} \to \Gamma (X, \widetilde{\mathcal M}) =: \widehat{\mathcal M}\). \({\iota}_ {\mathcal R}\) and \({\iota}_ {\mathcal M}\) are in general neither injective now surjective; in this paper the author studies conditions guaranteeing either injectivity or surjectivity or both, thus extending work of \textit{A. Buium} [Pac. J. Math. 98, 281-293 (1982; Zbl 0459.13015)], \textit{G. Carrà Ferro} [Rev. Roum. Math. Pures Appl. 30, 809-814 (1985; Zbl 0593.12016)], \textit{W. Keigher} [Pac. J. Math. 59, 99-122 (1975; Zbl 0327.12104); in: Contributions to Algebra: Coll. Papers Dedic. E. Kolchin, 239-249 (1977; Zbl 0369.12015); Adv. Math. 42, 143-153 (1981; Zbl 0479.14003); J. Algebra 79, 37-50 (1982; Zbl 0526.13018); J. Pure Appl. Algebra 27, 163-172 (1983; Zbl 0544.13018)] and the author [\textit{J. Kovacic}, in: Differential algebra and related topics, World Scientific, Singapore, 71-94 (2002; Zbl 1036.12004)]. Using the notion of \(\Delta\)-zero of a \(\Delta\)-\( \mathcal R\)-module, the author characterizes the kernel of \({\iota}_{\mathcal M}\): It is the \(\Delta\)-\({\mathcal R}\)-submodule \(\mathfrak Z({\mathcal M})\) consisting of all \(\Delta\)-zeroes of \(\mathcal M\). Even if \(\i…
reviewed by: Karl-Heinz Kiyek
Mathematics Subject Classification ID: 13N05
Mathematics Subject Classification ID: 14A15
Mathematics Subject Classification ID: 12H05
zbMATH DE Number: 1787453
zbMATH Keywords: differential ring
zbMATH Keywords: differential zero
zbMATH Keywords: differential unit
zbMATH Keywords: supersaturation
zbMATH Keywords: RAAD
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1787453