View source for Rigid analytic space

A variant of the concept of an [[Analytic space|analytic space]] related to the case where the ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r0823801.png" /> is a complete non-Archimedean normed field.

Analytic functions of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r0823802.png" />-adic variable were considered as long ago as the end of the 19th century in algebraic number theory, whereas the corresponding global object — a rigid analytic space — was introduced by J. Tate only in the early sixties of the 20th century (see [[#References|[1]]]). This construction was preceded by a more direct construction on the pattern of the theory of complex-analytic manifolds. The main lack of the latter approach is connected with the fact that the usual local definition of an analytic function as a power-series expansion in a neighbourhood of every point is inconvenient in view of the fact that the ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r0823803.png" /> is completely disconnected. The analytic functions defined in this way turned out to be  "too numerous"  (and, correspondingly, the analytic manifolds  "too few" ). For example, every compact analytic manifold over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r0823804.png" /> is the union of finitely many closed balls (see [[#References|[2]]]). Tate's construction starts with the local objects — the affinoid spaces, analogous to the affine varieties in algebraic geometry. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r0823805.png" /> be the algebra of power series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r0823806.png" /> variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r0823807.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r0823808.png" /> that converge in the polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r0823809.png" />. The quotient algebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238010.png" /> are called affinoid algebras. These algebras are Noetherian and they have a natural Banach topology in which all ideals are closed and all homomorphisms continuous. It turns out that every maximal ideal of such an algebra has finite codimension, and the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238011.png" /> of maximal ideals consists, up to conjugacy, of geometric points defined over finite extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238012.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238013.png" /> is the polydisc of unit radius, and, more generally, for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238014.png" /> the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238015.png" /> is an analytic subset (cf. [[Analytic set|Analytic set]]) of the polydisc. Homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238016.png" /> define morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238017.png" />, so that the affinoid spaces form a category.

A rigid structure on a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238018.png" /> is a collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238020.png" /> is a family of open sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238021.png" />, called admissible; for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238023.png" /> is a family of coverings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238024.png" /> by admissible sets (admissible coverings); and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238025.png" /> is a [[Pre-sheaf|pre-sheaf]] of rings on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238026.png" />. For an admissible covering it is required that certain natural axioms be satisfied, in particular, admissible coverings are refinable (cf. [[Refinement|Refinement]]), and the pre-sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238027.png" /> must be a [[Sheaf|sheaf]] with respect to all admissible coverings by admissible sets. Morphisms of spaces with a rigid structure, and also the concept of the rigid structure induced on a subspace, are defined by analogy with these concepts for ringed spaces. Every affinoid space may be endowed with a canonical rigid structure, which is preserved under morphisms. A rigid analytic space is, by definition, a topological space with a rigid structure on which there exists an admissible covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238028.png" /> such that every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238029.png" />, with the induced rigid structure, is isomorphic to an affinoid space equipped with the canonical rigid structure.

Several results have been obtained for rigid analytic spaces that are analogous to known theorems in the theory of complex spaces. Thus, there are analogues of Cartan's Theorems A and B (see [[Cartan theorem|Cartan theorem]], [[#References|[4]]]). More exactly, the coherent sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238030.png" />-modules on affinoid spaces are uniquely determined by the module of their sections and their cohomology spaces in dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238031.png" />. Also valid is the analogue of Grauert's theorem on the coherence of the image of a [[Coherent sheaf|coherent sheaf]] under a proper mapping (however, the definition of a proper mapping is very different from the usual one). A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082380/r08238032.png" />-adic analogue of [[Uniformization|uniformization]] of algebraic curves and algebraic varieties has been constructed (see ). A connection has been discovered between the concept of a rigid analytic space and that of a formal [[Scheme|scheme]] in algebraic geometry (see ).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Tate,   "Rigid analytic spaces"  ''Invent. Math.'' , '''12'''  (1971)  pp. 257–289</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.A. Bourbaki,   "Variétés différentielles et analytique. Fascicule de résultats" , ''Eléments de mathématiques'' , Hermann  (1967–1971)  pp. Fasc. XXXIII (Par. 1–7); Fasc. XXXVI (Par. 8–15)</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  J.-P. Serre,   "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  J.-P. Serre,   "Algèbres de Lie semi-simples complexes" , Benjamin  (1966)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  C. Housel,   "Espaces analytiques rigides" , ''Sem. Bourbaki (1966/67)'' , '''Exp. 32''' , Benjamin  (1968)</TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top">  D. Mumford,   "An analytic construction of degenerating curves over complete local rings"  ''Compos. Math.'' , '''24''' :  2  (1972)  pp. 129–174</TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top">  D. Mumford,   "An analytic construction of degenerating abelian varieties over complete rings"  ''Compos. Math.'' , '''24''' :  2  (1972)  pp. 239–272</TD></TR></table>