I am trying to show the following proposition:
Let $M$ be a differential manifold, $L$ be a line bundle, $D$ be a connection on $L$. Locally we can express $D=d+\theta$ for some one from $\theta$ (as $D(fe)=df~e+fDe=df~e+f\theta e$). Then the curvatrue form $\omega$ defined by $\omega(X,Y)=[D_X,D_Y]-D_{[X,Y]}$ is just $d\theta$.
My computation is as follows:
$ [D_X,D_Y]\\ =[X+\theta(X),Y+\theta(Y)]\\ =[X,Y]+X\theta(Y)-Y\theta(X)+\theta(X)Y-\theta(Y)X\\ =[X,Y]+d\theta(X,Y)+\theta(X)Y-\theta(Y)X ~(\text{use} ~d\theta(X,Y)=X\theta(Y)-Y\theta(X)) $
So remains to show $[X,Y]+\theta(X)Y-\theta(Y)X=D_{[X,Y]}=[X,Y]-\theta([X,Y])$, i.e. $\theta([X,Y])=\theta(X)Y-\theta(Y)X$. But this looks not likely to be true, since LHS is a constant and RHS is not. Where is the mistake?
And one relevant question: The curvature form $\omega$ defined by this way is equivalent to definition of $D^2e=\omega e$. How can I see this?
Thanks in advance.