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