### ${J}_{1}\left(p\right)$ has connected fibers.

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This paper lays the foundations for the global theory of irreducible components of rigid analytic spaces over a complete field $k$. We prove the excellence of the local rings on rigid spaces over $k$. This is used to prove the standard existence theorems and to show compatibility with the notion of irreducible components for schemes and formal schemes. Behavior with respect to extension of the base field is also studied. It is often necessary to augment scheme-theoretic techniques with other algebraic...

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...

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