Relative ampleness in rigid geometry (Q877536)

This is a foundational paper in rigid analysis. It concerns relative ampleness. The definition is: If \(f: X \rightarrow S\) is a proper rigid morphism and \(L\) a line bundle on \(X\), \(L\) is relatively ample over \(S\) if it is ample fiberwise. The author wants to link this definition to the algebro-geometrical one, and asks the following questions: (1) does there exist an admissible covering of \(S\) whose inverse image under \(f\) admits (locally over patches) an immersion into some projective space such that a positive power of \(L\) on this patch is isomorphic to the pull-back of \(O(1)\)? (2) Is the graded algebra \(\bigoplus f_* L^n\) locally finitely generated, and (3) can we recover \(X\) from this algebra? (The latter question having an affirmative answer in algebraic geometry by using relative Proj). The author establishes these results by reversing the usual logic. He uses formal scheme methods to first deduce a positive answer to question (2) and then uses an analytic relative Proj construction to answer (3), and only then question (1) is answered. This is due to the usual trouble with finding good admissible coverings from purely local properties.