Relative ampleness in rigid geometry

Brian Conrad[1]

  • [1] University of Michigan Department of Mathematics Ann Arbor, MI 48109 (USA)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 4, page 1049-1126
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Conrad, Brian. "Relative ampleness in rigid geometry." Annales de l’institut Fourier 56.4 (2006): 1049-1126. <http://eudml.org/doc/10167>.

@article{Conrad2006,
abstract = {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.},
affiliation = {University of Michigan Department of Mathematics Ann Arbor, MI 48109 (USA)},
author = {Conrad, Brian},
journal = {Annales de l’institut Fourier},
keywords = {Ampleness; rigid geometry; descent; ampleness},
language = {eng},
number = {4},
pages = {1049-1126},
publisher = {Association des Annales de l’institut Fourier},
title = {Relative ampleness in rigid geometry},
url = {http://eudml.org/doc/10167},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Conrad, Brian
TI - Relative ampleness in rigid geometry
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 4
SP - 1049
EP - 1126
AB - 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.
LA - eng
KW - Ampleness; rigid geometry; descent; ampleness
UR - http://eudml.org/doc/10167
ER -

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