nLab non-archimedean analytic geometry




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 KK 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 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 and, 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:


For literature on specific approaches see the nnLab 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 pp-adic geometry – From Hensel to Berkovich and beyond, talk notes, June 2012 (pdf)

A standard textbook on rigid analytic geometry is

For comparison see

category: geometry

Last revised on March 2, 2024 at 06:08:13. See the history of this page for a list of all contributions to it.