Differential étale extensions and differential modules over differential rings
type: Item
instance of: scholarly article
title: Differential étale extensions and differential modules over differential rings
zbMATH Open document ID: 1362.12003
published in: Advances in Applied Mathematics
publication date: 2015-12-11T00:00:00Z
review text: An extension of rings \(R/F\) is defined to be formally étale if it satisfies the following lifting property: For all surjective homomorphisms of \(F\)-algebras \(f:A\to S\) whose kernel \(I=\mathrm {ker}(f)\) satisfies \(I^2=0\) and all \(F\)-algebra homomorphisms \(g: R\to S\) there is a lifting of \(g\) to \(h: R\to A\), i.e. an \(F\)-algebra homomorphism \(h: R\to A\) such that \(f\circ h=g\). The author is concerned with a similar property for differential ring extensions which might be called ``differentially étale'', and which is obtained by replacing rings by differential rings and homomorphisms by differential homomorphisms. One gets a necessary condition for such an extension to be differentially étale by taking the special case of \(R=S\) and \(g=\mathrm{id}_R\), in which the lifting property states that any such surjection \(A\to R\) admits a differential splitting \(h: R\to A\). The author therefore considers surjections \(f: A\to R\) with \(\mathrm {ker}(f)^2=0\) which he calls ``differential square zero extensions of \(R\)''. For a differentially simple ring \(R/F\) and a differential module \(I\) over \(R\) (fulfilling some technical assumptions), the main theorem gives equivalent conditions to when all differential square zero extensions \(A\to R\) with kernel \(I\) are differentially split. As a main corollary, he obtains that for the maximal Picard-Vessiot ring \(R\) over \(F\) (i.e. the maximal differential ring extension of \(F\) such that all elements of \(R\) are solutions of some linear differential equation over \(F\)), and any ideal \(I\) of \(R\), all differential square zero extensions \(A\to R\) with kernel \(I\) are differentially split.
reviewed by: Andreas Maurischat
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 13N05
zbMATH DE Number: 6519863
zbMATH Keywords: differential rings
zbMATH Keywords: differential Galois
zbMATH Keywords: Picard-Vessiot
author: Andy R. Magid
MaRDI profile type: Publication
full work available at URL: j.aam.2015.09.006
OpenAlex ID: W2212534358
cites work: Solution algebras of differential equations and quasi-homogeneous varieties: a new differential Galois correspondence
cites work: Projectivity of modules over rings with suitable group action
cites work: Cohomology of rings with algebraic group action
cites work: Equivariant completions of rings with reductive group action
cites work: Q4663641
cites work: Q5611879
cites work: Q3866302
cites work: Q4039952
cites work: Q4790565
DOI: 10.1016/J.AAM.2015.09.006
description: scientific article; zbMATH DE number 6519863