Principal parts on the projective line over arbitrary rings

Abstract

We develop techniques to split explicitly the sheaf of principal parts ${\mathcal P^k(\mathcal O(n))}$ as left and right module on the projective line over an arbitrary ring. We then apply the techniques developed to split the principal parts ${\mathcal P^k(\mathcal O(n))}$ for all ${n\,\in {\bf Z}}$ and k ≥ 1 as left and right ${\mathcal O}$-module on the projective line over any field of characteristic zero giving a complete description of the principal parts on the projective line.