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

Published online by Cambridge University Press:  23 August 2005

L. Lipshitz
Affiliation:
Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA. E-mail: lipshitz@math.purdue.edu
Z. Robinson
Affiliation:
Department of Mathematics, East Carolina University, Greenville, NC 27858, USA. E-mail: robinsonz@mail.ecu.edu
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Abstract

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 of Hironaka's complex analytic flattening theorem given by T. Gardener and H. Schoutens, in Theorem 2.3 of ‘Flattening and subanalytic sets in rigid analytic geometry’, Proc. London Math. Soc (3) 83 (2001) 681–707, is not true. Since this is a key step in the proof of the affinoid elimination theorem (loc. cit. 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.

Type
Research Article
Copyright
2005 London Mathematical Society

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Footnotes

This research was supported in part by NSF grant number DMS 0070724.