The Cyclic Vector Lemma

TL;DR

This paper explores the conditions under which a differential module over a differential field is singly generated, linking it to the existence of certain module injections related to the differential Galois group.

Contribution

It establishes a criterion for when a differential module is singly generated based on the existence of a module injection involving the Picard--Vessiot ring and the Galois group.

Findings

A differential module is singly generated iff a specific module injection exists.

Such an injection always exists if the constant field is algebraically closed and different from the base field.

Provides a characterization connecting module generation to Galois group representations.

Abstract

Let $F$ be a differential field of characteristic zero with algebraically closed constant field $C$. Let $E$ be a Picard--Vessiot closure of $F$, $R \subset E$ its Picard--Vessiot ring and $\Pi$ the differential Galois group of $E$ over $F$. Let $V$ be a differential $F$ module, finite dimensional as an $F$ vector space. Then $V$ is singly generated as a differential $F$ module if and only if there is a $\Pi$ module injection $\text{Hom}_F^\text{diff}(V,R) \to R$. If $C \neq F$ such an injection always exists.