nLab
non-archimedean analytic geometry

Idea

Non-archimedean geometry is geometry over non-archimedean fields. While the concrete results are quite different, the basic formalism of algebraic schemes and formal schemes over a non-archimedean field KK is the special case of the standard formalism over any field. The “correct” analytic geometry over non-archimedean field, however, is not a straightforward analogue of the complex analytic case. As Tate noticed, the sheaf of KK-valued functions which can be locally written as converging power series over the affine space K nK^n is too big (too many analytic functions) due to the fact that the underlying topological space is totally disconnected. Also there are very few KK-analytic manifolds. This naive approach paralleling the complex analytic geometry is called by Tate wobbly KK-analytic varieties an, apart from the case of non-archimedean local fields it is of little use. For this reason Tate introduced a better KK-algebra of analytic functions, locally takes its maximal spectrum and made a Grothendieck topology which takes into account just a certain smaller set of open covers; this topology is viewed as rigidified, hence the varieties based on gluing in this approach is called rigid analytic geometry. Raynaud has shown how some classes of rigid KK-varieties can be realized as generic fibers of formal schemes over the ring of integers of KK; this is called a formal model of a rigid variety. Different formal models are birationally equivalent, more precisely they are related via admissible blow-ups. Later more sophisticated approaches appeared:

Literature

For literature on specific approaches see the nnLab entries Berkovich analytic space, adic space, global analytic geometry, rigid analytic geometry. For comparison see

  • Brian Conrad, Several approaches to non-archimedean geometry, lectures at Arizona winter school 2007, pdf

  • Kiran S. Kedlaya, Reified valuations and (re)adic spectra, arxiv/1309.0574

  • Kazuhiro Fujiwara, Fumiharu Kato, Foundations of Rigid Geometry I, arxiv/1308.4734

category: geometry

Revised on September 4, 2013 16:28:04 by David Corfield (87.114.166.212)