Dear @JasonStarr, I am confused. I've looked it up in the Serre`s book and have not find information concerning this question. To this point I am not sure that I formulated it clear enough. My question is the following. For any ring $R$ an element in $PGL(R((h)))$ defines a line bundle $L$ on $Spec(R((h)))$. I wonder whether it is true that there exists an open covering $Spec R = \cup Spec R_{f_i}$ such that pullbacks of $L$ to $Spec(R_{f_i}((h)))$ are trivial. I don't see why your ring provides a counterexample, so I will be grateful if you explain it in more details.

@JasonStarr thanks a lot! But could you be a little more specific? In which chapter can I find this discussion?