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 $K$ 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 $K$-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 an, 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:

Literature

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

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

category: geometry

Revised on May 9, 2013 11:49:50 by David Roberts (192.43.227.18)