Twisted Poincar\'e Lemma and Twisted \v{C}ech-de Rham Isomorphism in case of Projective Line

TL;DR

This paper provides a direct proof of the twisted Poincaré lemma using integration over regularized paths, offering a concrete description of the Čech-de Rham isomorphism for the projective line.

Contribution

It introduces a new proof technique for the twisted Poincaré lemma and clarifies the Čech-de Rham isomorphism in the context of the projective line.

Findings

Concrete description of the Čech-de Rham isomorphism

Direct proof of the twisted Poincaré lemma using regularized paths

Enhanced understanding of the twisted Poincaré lemma in algebraic geometry

Abstract

In this paper, we give a direct proof of the twisted Poincar\'{e} lemma by using the integrations over regularized paths. This method tells us a concrete description of the \v{C}ech-de Rham isomorphism.