Rigid analytic space
Encyclopedia
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 curve
s 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.
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 ideal
s 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.
R of k. In particular, he showed that the category of quasi-compact quasi-separated rigid spaces over k is equivalent to the localization of the category of quasi-compact admissible formal schemes over R with respect to admissible formal blow-ups. Here, a formal scheme is admissible if it is coverable by formal spectra of topologically finitely presented R algebras whose local rings are R-flat.
Formal models suffer from a problem of uniqueness, since blow-ups allow more than one formal scheme to describe the same rigid space. Huber worked out a theory of adic spaces to resolve this, by taking a limit over all blow-ups. These spaces are quasi-compact, quasi-separated, and functorial in the rigid space, but lack a lot of nice topological properties.
Berkovich reformulated much of the theory of rigid analytic spaces in the late 1980s, using a generalization of the notion of Gelfand spectrum for commutative unital C*-algebras. The Berkovich spectrum of a Banach k-algebra A is the set of semi-norms on A that are bounded with respect to the given norm on k, and it has a topology induced by evaluating these semi-norms on elements of A. Since the topology is pulled back from the real line, Berkovich spectra have many nice properties, such as compactness, path-connectedness, and metrizability. Many ring-theoretic properties are reflected in the topology of spectra, e.g., if A is Dedekind, then its spectrum is contractible. However, even very basic spaces tend to be unwieldy – the projective line over Cp is a compactification of the inductive limit of affine Bruhat–Tits buildings for PGL2(F), as F varies over finite extensions of Qp, when the buildings are given a suitably coarse topology.
Complex analytic space
In mathematics, a complex analytic space is a generalization of a complex manifold which allows the presence of singularities. Complex analytic spaces are locally ringed spaces which are locally isomorphic to local model spaces, where a local model space is an open subset of the vanishing locus of...
over a nonarchimedean field. They were introduced by John Tate
John Tate
John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.-Biography:...
in 1962, as an outgrowth of his work on uniformizing p-adic elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s with bad reduction using the multiplicative group
Multiplicative group
In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the context*any group \scriptstyle\mathfrak \,\! whose binary operation is written in multiplicative notation ,*the underlying group under multiplication of the invertible elements of...
. In contrast to the classical theory of p-adic analytic manifolds
P-adic analysis
In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers....
, rigid analytic spaces admit meaningful notions of analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
and connectedness
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
. 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 ringPolynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
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
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. Points on the polydisc are defined to be maximal ideal
Maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
s in the Tate algebra, and if k is algebraically closed
Algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
, 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
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
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
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, but they do form a Grothendieck topology
Grothendieck topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...
(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 schemesScheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
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.
Other formulations
Around 1970, Raynaud provided an interpretation of certain rigid analytic spaces as formal models, i.e., as generic fibers of formal schemes over the valuation ringValuation ring
In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D....
R of k. In particular, he showed that the category of quasi-compact quasi-separated rigid spaces over k is equivalent to the localization of the category of quasi-compact admissible formal schemes over R with respect to admissible formal blow-ups. Here, a formal scheme is admissible if it is coverable by formal spectra of topologically finitely presented R algebras whose local rings are R-flat.
Formal models suffer from a problem of uniqueness, since blow-ups allow more than one formal scheme to describe the same rigid space. Huber worked out a theory of adic spaces to resolve this, by taking a limit over all blow-ups. These spaces are quasi-compact, quasi-separated, and functorial in the rigid space, but lack a lot of nice topological properties.
Berkovich reformulated much of the theory of rigid analytic spaces in the late 1980s, using a generalization of the notion of Gelfand spectrum for commutative unital C*-algebras. The Berkovich spectrum of a Banach k-algebra A is the set of semi-norms on A that are bounded with respect to the given norm on k, and it has a topology induced by evaluating these semi-norms on elements of A. Since the topology is pulled back from the real line, Berkovich spectra have many nice properties, such as compactness, path-connectedness, and metrizability. Many ring-theoretic properties are reflected in the topology of spectra, e.g., if A is Dedekind, then its spectrum is contractible. However, even very basic spaces tend to be unwieldy – the projective line over Cp is a compactification of the inductive limit of affine Bruhat–Tits buildings for PGL2(F), as F varies over finite extensions of Qp, when the buildings are given a suitably coarse topology.