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rigid analytic space
In mathematics, a rigid analytic space is an analogue of a complex analytic space over a nonarchimedean field. They were introduced by John Tate in 1962, as an outgrowth of his work on uniformizing ''p''-adic elliptic curves with bad reduction using the multiplicative group. In contrast to the classical theory of ''p''-adic analytic manifolds, rigid analytic spaces admit meaningful notions of analytic continuation and connectedness. However, this comes at the cost of some conceptual complexity.
==Definitions==
The basic rigid analytic object is the ''n''-dimensional unit polydisc, whose ring of functions is the Tate algebra ''Tn'', made of power series in ''n'' variables whose coefficients approach zero in some complete nonarchimedean field ''k''. The Tate algebra is the completion of the polynomial ring in ''n'' variables under the Gauss norm (taking the supremum of coefficients), and the polydisc plays a role analogous to that of affine ''n''-space in algebraic geometry. Points on the polydisc are defined to be maximal ideals in the Tate algebra, and if ''k'' is algebraically closed, these correspond to points in ''kn'' whose coordinates have size at most one.
An affinoid algebra is a ''k''-Banach algebra that is isomorphic to a quotient of the Tate algebra by an ideal. An affinoid is then a subset of the unit polydisc on which the elements of this ideal vanish, i.e., it is the set of maximal ideals containing the ideal in question. The topology on affinoids is subtle, using notions of ''affinoid subdomains'' (which satisfy a universality property with respect to maps of affinoid algebras) and ''admissible open sets'' (which satisfy a finiteness condition for covers by affinoid subdomains). In fact, the admissible opens in an affinoid do not in general endow it with the structure of a topological space, but they do form a Grothendieck topology (called the ''G''-topology), and this allows one to define good notions of sheaves and gluing of spaces.
A rigid-analytic space over ''k'' is a pair describing a locally ringed ''G''-topologized space with a sheaf of ''k''-algebras, such that there is a covering by open subspaces isomorphic to affinoids. This is analogous to the notion of manifolds being coverable by open subsets isomorphic to euclidean space, or schemes being coverable by affines. Schemes over ''k'' can be analytified functorially, much like varieties over the complex numbers can be viewed as complex analytic spaces, and there is an analogous formal GAGA theorem. The analytification functor respects finite limits.
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