Inheritance of properties of spectra
type: Item
instance of: scholarly article
title: Inheritance of properties of spectra
zbMATH Open document ID: 1236.13025
DOI: 10.1134/S0001434610110271
published in: Mathematical Notes
publication date: 2011-11-25T00:00:00Z
review text: 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}, …
reviewed by: Oleksandr Iena
Mathematics Subject Classification ID: 13N99
Mathematics Subject Classification ID: 14Axx
zbMATH DE Number: 5980731
zbMATH Keywords: inheritance of properties of spectra
zbMATH Keywords: differential algebraic variety
zbMATH Keywords: prime differential ideal
zbMATH Keywords: differential spectrum of a ring
zbMATH Keywords: Keigher ring
zbMATH Keywords: morphism of differential algebraic varieties
zbMATH Keywords: differentially finitely generated algebra
zbMATH Keywords: dominant morphism
zbMATH Keywords: open map
author: Dmitry Trushin
MaRDI profile type: Publication
cites work: Differential Algebraic Groups
cites work: Q5680222
cites work: Q3220670
cites work: Q5563439
cites work: Q3255437
full work available at URL: s0001434610110271
OpenAlex ID: W2039880672
description: scientific article; zbMATH DE number 5980731