Let X be a complex manifold. Suppose we have holomorphic line bundles $L_i$ over $U_i$ where ${U_i}$ is an open covering of X. Suppose that $L_i$ and $L_j$ restrict to the same line bundle over the intersection of $U_i$ and $U_j$.

Can we patch these local line bundles into a global holomorphic line bundle L over X? That is, the restriction of L to $U_i$ is $L_i$.

equal, then yes, nothing could possibly go wrong. But that situation rarely (ever?) arises in practice. Normally what you have is that the bundles areisomorphicon the intersections, and then indeed an additional condition is needed, namely compatibility on the triple intersections (cocycle condition). Still, this is explained everywhere line bundles are peddled, so I find the question somewhat mysterious. – Pete L. Clark Sep 3 '10 at 9:46