Differential étale extensions and differential modules over differential rings (Q895976)

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.