Flattening and analytic continuation of affinoid morphisms: remarks on a paper of Gardener and Schoutens

We give an example of an affinoid curve without analytic continuation. We use this to produce an example of an affinoid morphism that cannot be flattened by a finite sequence of local blow-ups. Thus the global rigid analogue, [7], Theorem 2.3, of Hironaka’s complex analytic flattening theorem is not true. Since this is a key step in the proof of the affinoid elimination theorem, [7], Theorem 3.12, that proof contains a serious gap. We also give an example of an affinoid subset of the plane that is not the image under a proper rigid analytic map of a set that is globally semianalytic in the domain of that map. This clarifies the relationship among several natural categories of rigid subanalytic sets.