Relative Ampleness in Rigid Geometry

We develop a rigid-analytic theory of relative ampleness for line bundles and record some applications to faithfully flat descent for morphisms and proper geometric objects. The basic definition is fibral, but pointwise arguments from the algebraic and complex-analytic cases do not apply, so we use cohomological properties of formal schemes over completions of local rings on rigid spaces. An analytic notion of quasi-coherence is introduced so that we can recover a proper object from sections of an ample bundle via suitable Proj construction. The locus of relative ampleness in the base is studied, as is the behavior of relative ampleness with respect to analytification and arbitrary extension of the base field. In particular, we obtain a quick new proof of the relative GAGA theorem over affinoids.