Let $X$ be an integral scheme with function field $K$. Let $\mathcal{L}$ be an invertible sheaf on $X$, and let $\mathcal{L}_K = \mathcal{L} \otimes_{\mathcal{O}_X} K$, where $K$ is the constant sheaf on $X$ with value $K$.
Locally $\mathcal{L}_K|_U \simeq K|_U$, but do we have a global isomorphism $\mathcal{L}_K \cong K$?
Also, is $H^1(X, \mathcal{L}_K)=0$? Assume $X$ is separated and whatever else is needed for Cech cohomology to compute sheaf cohomomology.