Rigid analytic geometry
Tate m´a écrit de son côté sur ses histoires de courbes elliptiques, et pour me demander si j´avais des idées sure une définition globale des variétés analytiques sur des corps complets. Je dois avouer que je n´ai pas du tout compris pourquoi ses résultats suggéreraient l´existence d´une telle définition, et suis encore sceptique.
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.
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 norm 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.
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