In Huybrechts' Complex Geometry 4.4 Chern Classes Proposition 4.4.12,let L be a complex line bundle over a differentiable manifold $M$.Then the image of $\delta (L)\in H^2(M,\mathbb Z)$ under the natural map $H^2(M,\mathbb Z)\subset H^2(M,\mathbb C)$ equals $-c_1(L)$ ,where $\delta $ is the boundary isomorphism: $H^1(M,\mathcal C^*_{\mathbb C}) \cong H^2(M,\mathbb Z)$.Then the proof is:
What make me confused is the underlined word in the picture,isn't $\varphi_{jk}-\varphi _{ik}+\varphi_{ij}=0?$If it is,it surely induces zero class in $H^2({U_i},\mathbb Z)$,then determines zero class in $H^2(M,\mathbb Z)=\lim_{\rightarrow}H^2({U_i},\mathbb Z)$ that must be wrong.So where did I miss?Any help is appreciated.Thanks in advance!
It’s not necessarily 0 for the same reason that $\exp(a+b) =\exp(c)$ doesnt imply $a+b=c$ (only up to an integer).