Euler Sequence and Koszul complex of a module

TL;DR

This paper develops Euler sequences for modules, uses them to analyze Koszul complexes, and generalizes Bott's formula to projective bundles over schemes of characteristic zero.

Contribution

It introduces relative and global Euler sequences for modules and applies these to extend Bott's formula to a broader class of schemes.

Findings

Proved acyclicity results for Koszul complexes

Computed cohomology of differential sheaves on projective bundles

Generalized Bott's formula to schemes of characteristic zero

Abstract

We construct relative and global Euler sequences of a module. We apply it to prove some acyclicity results of the Koszul complex of a module and to compute the cohomology of the sheaves of (relative and absolute) differential $p$-forms of a projective bundle. In particular we generalize Bott's formula for the projective space to a projective bundle over a scheme of characteristic zero.