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