Vector fields and differential schemes

TL;DR

This paper extends the concepts of vector fields and trajectories to schemes, providing a geometric interpretation and generalizing differential Galois theory results, with applications to sheaves over the differential spectrum.

Contribution

It introduces a framework for vector fields on schemes and generalizes key results in differential Galois theory and differential schemes.

Findings

Generalized differential Galois theory results

Proved a theorem on extension of constant sections

Compared classical sheaves over the differential spectrum

Abstract

We define vector fields, leaves and trajectories for schemes. With these tools, we are able to give a geometrical interpretation and to generalize several results of differential Galois theory and constructions on differential schemes. We prove a theorem of extension of constant sections. Finally, as an application, we compare three classical sheaves defined over the differential spectrum.