Higher order differential operators on projective modules

TL;DR

This paper provides explicit formulas for higher order differential operators on finitely generated projective modules over commutative rings, and explores their applications to curvature and stratification concepts.

Contribution

It introduces explicit formulas for higher order differential operators and applies them to compute curvature and analyze stratifications on projective modules.

Findings

Explicit formulas for higher order differential operators on projective modules

A simple formula for the curvature of a connection using a projective basis

Few stratifications are induced by a projective basis

Abstract

In this paper we give explicit formulas for higher order differential operators on a finitely generated projective module $E$ on an arbitrary commutative unital ring $A$. We use the differential operators constructed to give a simple formula for the curvature of a classical connection and a connection on a Lie-Rinehart algebra in terms of a "projective basis" $B$ for $E$. A "projective basis" is sometimes referred to as a "dual basis". This gives an explicit formula for the curvature $R_{\nabla_B}$ of a connection $\nabla_B$ on $E$ defined in terms of a projective basis $B$ and an idempotent $\phi$ for $E$. We also consider the notion of a stratification on the module $E$ induced by a projective basis $B$. It turns out few stratifications are induced by a projective basis.