nLab non-archimedean analytic geometry

Idea

Non-archimedean geometry is (algebraic) 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 $K$ is the special case of the standard formalism over any field. The “correct” analytic geometry over non-archimedean fields, however, is not a straightforward analogue of the complex analytic case.

As Tate noticed, the sheaf of $K$-valued functions which can be locally written as converging power series over the affine space $K^n$ is too big (too many analytic functions) due to the fact that the underlying topological space is totally disconnected. Also there are very few $K$-analytic manifolds. This naive approach paralleling the complex analytic geometry is called by Tate wobbly $K$-analytic varieties and, apart from the case of non-archimedean local fields it is of little use. For this reason Tate introduced a better $K$-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 $K$-varieties can be realized as generic fibers of formal schemes over the ring of integers of $K$; 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:

• Berkovich analytic spaces, which take prime spectra together with an extension of the abolsute value to the residue field
• Huber’s adic spaces which take into account also valuations of higher rank
• Fujiwara-Kato’s approach (‘rigid geometry’) which takes a limit over all formal models; this has spirit of birational geometry
• global analytic geometry.

Literature

For literature on specific approaches see the $n$Lab entries Berkovich analytic space, adic space, global analytic geometry, rigid analytic geometry, Huber space, perfectoid space.

A nice survey is in

• Bernard Le Stum, One century of $p$-adic geometry – From Hensel to Berkovich and beyond, talk notes, June 2012 (pdf)

A standard textbook on rigid analytic geometry is

• Siegfried Bosch, Ulrich Güntzer, Reinhold Remmert, Non-Archimedean Analysis – A systematic approach to rigid analytic geometry, Grundlehren der Mathem. Wissen. 261 (1984) [ ISBN:9783540125464, pdf]

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

• Oren Ben-Bassat, Kobi Kremnizer, Non-Archimedean analytic geometry as relative algebraic geometry (arXiv:1312.0338)

• S. Bosch, U. Güntzer, Reinhold Remmert, Non-Archimedean Analysis – A systematic approach to rigid analytic geometry, 1984 (pdf)

category: geometry