Global sections of diffspec
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 \(\iota_{\mathcal M}: {\mathcal M} \to \widehat{\mathcal M}\) is injective, this need not to be true after localization. To ensure that, the author introduces the condition RAAD (= ``radical annihilators are differential'') [cf. proposition 6.2 for a number of equivalent conditions]. Surjectivity of the map \(\iota_{\mathcal R}: {\mathcal R} \to \widehat{\mathcal R}\) is studied in sections 9-12. For a multiplicatively closed subset \(S \subset {\mathcal R}\) the author introduces the notion of \(\Delta\)-supersaturation; one of the main results is the following: If the set \({\mathcal U}\) of \(\Delta\)-units of \({\mathcal R}\) is \(\Delta\)-supersaturated and \({\mathcal M}\) satisfies condition RAAD, then \({\mathcal U}^{- 1} {\mathcal M} \to \widehat{\mathcal M}\) is an isomorphism. In the last sections the author uses the fact that the functor \(\mathfrak Z: \text{mod}_{\mathcal R}\to\text{mod}_{\mathcal R}\) is an idempotent radical to equip \({\mathcal R}\) and \({\mathcal M}\) with a Gabriel topology \(\mathfrak G\), thus extending results of Buium [loc. cit.]. We mention one result: If \({\mathcal M}\) is RAADZ (= ``radical annihilators are differential zeros'') [cf. proposition 7.1], then the canonical mapping \(\varphi_{\mathcal M} : {\mathcal Q} ({\mathcal M}) \to \widehat{\mathcal M}\) is a \(\Delta\)-isomorphism [here \({\mathcal Q} ({\mathcal M})\) denotes the module of quotients with respect to the Gabriel topology \(\mathfrak G\) of \({\mathcal M}\)]. Let \(\mathfrak G^1\) be the set of ideals \(I \in \mathfrak G\) with \(I \cap {\mathcal U} \neq \emptyset\). Then \(\mathfrak G^1\) is a principal Gabriel topology; if \({\mathcal M}\) is RAADZ, then one has an exact sequence \[ \to {\mathcal U}^{- 1} \mathfrak Z ({\mathcal M}) \to {\mathcal U}^{- 1} {\mathcal M} \to \widehat{\mathcal M} \to {\mathcal Q} ({\mathcal M}) / {\mathcal Q}^1 ({\mathcal M}) \to 0. \] The paper contains numerous examples illustrating the concepts introduced by the author.
scientific article; zbMATH DE number 1787453
scientific article