Inheritance of properties of spectra
The paper under review presents a technique, called inheritance, that allows to derive some properties of differential spectra of differential rings from the corresponding properties of commutative spectra. The technique is applied to prove certain geometric properties of morphisms of differential algebraic varieties. All the rings in the paper are associative commutative rings with identity. A differential ring is a ring with a fixed set of derivations. A differential ideal is an ideal stable under the derivations. A differential prime ideal is a prime ideal stable under the derivations. This way one defines differential spectra of differential rings. Given a fixed bijection between the sets of derivations of two differential rings, a homomorphism of differential rings is then a homomorphism of rings commuting with the derivations. A Ritt algebra is a differential ring that is an algebra over the field of rational numbers \(\mathbb Q\). In Statement~2 the author shows that differential spectra inherit some important properties of usual spectra. In particular the going-up and the going-down properties are inherited by differential spectra. Statement~5 is an important technical result used in the paper. It says that for a differentially finitely generated integral algebra \(B\) over a Ritt algebra \(A\) there exist an element \(s\) in \(B\), finitely generated algebra \(C\) over \(A\), and at most countable set of variables \(\{y_\alpha\}\) algebraically independent over \(C\) such that the localization \(B_s=C[y_{\alpha}]\). The main result of the paper is Theorem~15. It states that given an inclusion of differential Ritt algebras \(A\subset B\) such that \(B\) is differentially finitely generated over \(A\), there exists an element \(s\) of \(B\) such that the inclusion \(A\subset B_s\) has the going-down property for differential ideals. The proof of Theorem~15 uses Statement~5 along with statements from commutative algebra from the books [\textit{H.~ Matsumura}, Commutative algebra. 2nd ed. Mathematics Lecture Note Series, 56. Reading, Massachusetts, etc.: The Benjamin/Cummings Publishing Company, Inc., Advanced Book Program (1980; Zbl 0441.13001)] and [\textit{M. F.~Atiyah}, \textit{I.~G.~Macdonald}, Introduction to commutative algebra. Reading, Mass.-Menlo Park, Calif.- London-Don Mills, Ont.: Addison- Wesley Publishing Company (1969; Zbl 0175.03601)], in particular going-down properties. Statement~2 insures that the going-down property is inherited by differential ideals. The paper under review consists of 11 sections. Section~1 is an introduction. In Section~2 the structure of the paper is described. In Section~3 the author provides some definitions and notations and refers to the books [\textit{E.~R.~Kolchin}, Differential algebra and algebraic groups. Pure and Applied Mathematics, 54. New York-London: Academic Press. (1973; Zbl 0264.12102)] and [\textit{M. F.~Atiyah}, \textit{I.~G.~Macdonald}, Introduction to commutative algebra. Reading, Mass.-Menlo Park, Calif.- London-Don Mills, Ont.: Addison- Wesley Publishing Company (1969; Zbl 0175.03601)] for notations and definitions used but not mentioned explicitly in the paper. The main method and basic statements concerning the inheritance of properties are presented in Section~4, their direct corollaries are given in Section~5. A technical result, Statement~5, is proved in Section~6. Statement~5 is applied in Section~7 in order to prove some known results. In Section~8 algebraic results are translated into the language of spectra. The main theorem is proved in Section~9. Section~10 deals with locally closed points of differential spectra, which play a similar role to that of maximal spectra in algebraic geometry. Section~11 presents some geometric statements about differential algebraic varieties and groups.
morphism of differential algebraic varieties
differentially finitely generated algebra
scientific article; zbMATH DE number 5980731