Berkovich Space
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Berkovich space, introduced by , is a version of an analytic space over a non-Archimedean field (e.g. ''p''-adic field), refining Tate's notion of a rigid analytic space.


Motivation

In the
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
case,
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
begins by defining the complex affine space to be \Complex^n. For each U\subset\Complex^n, we define \mathcal_U, the
ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...
of
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s on U to be the ring of
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s, i.e. functions on U that can be written as a convergent
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
in a
neighborhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
of each point. We then define a local model space for f_, \ldots, f_\in\mathcal_U to be :X:=\ with \mathcal_X=\mathcal_U/(f_, \ldots,f_). A
complex analytic space In mathematics, particularly differential geometry and complex geometry, a complex analytic varietyComplex analytic variety (or just variety) is sometimes required to be irreducible and (or) Reduced ring, reduced or complex analytic space is a g ...
is a locally ringed \Complex-space (Y, \mathcal_Y) which is locally isomorphic to a local model space. When k is a
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
non-Archimedean field, we have that k is
totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
. In such a case, if we continue with the same definition as in the complex case, we wouldn't get a good analytic theory. Berkovich gave a definition which gives nice analytic spaces over such k, and also gives back the usual definition over \Complex. In addition to defining analytic functions over non-Archimedean fields, Berkovich spaces also have a nice underlying
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
.


Berkovich spectrum

A
seminorm In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
on a ring A is a non-constant function , \!-\!, : A \to \R_ such that :\begin , 0, &=0 \\ , 1, &=1 \\ , f+g, &\leqslant , f, +, g, \\ , fg, &\leqslant , f, , g, \end for all f, g \in A. It is called multiplicative if , fg, =, f, , g, and is called a norm if , f, = 0 implies f = 0. If A is a normed ring with norm \, \!-\!\, then the Berkovich spectrum of A, denoted \mathcal(A), is the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of multiplicative seminorms on A that are bounded by the norm of A. The Berkovich spectrum is equipped with the weakest
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
such that for any f \in A the map :\begin \varphi_:\mathcal(A)\to\R \\ , \cdot, \mapsto , f, \end is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
. The Berkovich spectrum of a normed ring A is
non-empty In mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, whil ...
if A is non-zero and is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
if A is complete. If x is a point of the spectrum of A then the elements f with , f, _x = 0 form a
prime ideal In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...
of A. The
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the fie ...
of the quotient by this prime ideal is a normed field, whose completion is a complete field with a multiplicative norm; this field is denoted by \mathcal(x) and the image of an element f\in A is denoted by f(x). The field \mathcal(x) is generated by the image of A. Conversely a bounded map from A to a complete normed field with a multiplicative norm that is generated by the image of A gives a point in the spectrum of A. The spectral radius of f, :\rho(f)=\lim_ \left \, f^n \right \, ^ is equal to :\sup_, f, _.


Examples

* The spectrum of a field complete with respect to a valuation is a single point corresponding to its valuation. * If A is a commutative C*-algebra then the Berkovich spectrum is the same as the
Gelfand spectrum In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things: * a way of representing commutative Banach algebras as algebras of continuous functions; * the fact that for commutative C*-alg ...
. A point of the Gelfand spectrum is essentially a
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
to \Complex, and its absolute value is the corresponding seminorm in the Berkovich spectrum. *
Ostrowski's theorem In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers \Q is equivalent to either the usual real absolute value or a -adic absolute value. Definitions An a ...
shows that any multiplicative seminorm on the
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s (with the usual absolute value) is one of the following four types: , -, _ (0 < \varepsilon \leq 1), , -, _0, , -, _ (0 < \varepsilon < 1) and , -, _ (p a prime). Here , -, _ = , -, _^\varepsilon, , -, _ is the p-adic norm for which , p, = \varepsilon, , -, _ is the seminorm induced by the trivial norm , -, _ on \mathbb/p, and , -, _ is the trivial norm on \mathbb, i.e., the norm which sends all nonzero elements to 1. For each p (prime or infinity) we get a branch which is
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to a real interval, the branches meet at the point corresponding to the trivial valuation. The open neighborhoods of the trivial valuations are such that they contain all but finitely many branches, and their intersection with each branch is open.


Berkovich affine space

If k is a field with a valuation, then the ''n''-dimensional Berkovich affine space over k, denoted \mathbb^n_k, is the set of multiplicative seminorms on k _1, \ldots,x_n/math> extending the norm on k. The Berkovich affine space is equipped with the weakest topology such that for any f\in k the map \varphi_: \mathbb^n \to\R taking , \cdot, \in\mathbb^n to , f, is continuous. This is not a Berkovich spectrum, but is an increasing union of the Berkovich spectra of rings of power series that converge in some ball (so it is locally compact). We define an analytic function on an open subset U\subset\mathbb^ as a map :f:U\to\prod_\mathcal(x) with f(x)\in\mathcal(x), which is a local limit of rational functions, i.e., such that every point x\in U has an open neighborhood U'\subset U with the following property: : \forall \varepsilon > 0\,\exist g, h \in\mathcal _, \ldots, x_ \qquad \forall x' \in U' \left( h(x') \neq 0 \ \,\land\ \left, f(x')-\frac\ < \varepsilon \right). Continuing with the same definitions as in the complex case, one can define the ring of analytic functions, local model space, and analytic spaces over any field with a valuation (one can also define similar objects over normed rings). This gives reasonable objects for fields complete with respect to a nontrivial valuation and the ring of integers \Z. In the case where k=\Complex, this will give the same objects as described in the motivation section. These analytic spaces are not all analytic spaces over non-Archimedean fields.


Berkovich affine line

The 1-dimensional Berkovich affine space is called the Berkovich affine line. When k is an
algebraically closed In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra h ...
non-Archimedean field, complete with respects to its valuation, one can describe all the points of the affine line. There is a canonical
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup. When some object X is said to be embedded in another object Y ...
k\hookrightarrow\mathbb^1_k . The space \mathbb^ is a locally compact, Hausdorff, and uniquely
path-connected 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 non-empty open subsets. Connectedness is one of the principal topological properties t ...
topological space which contains k as a
dense Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...
subspace. One can also define the Berkovich projective line \mathbb^ by adjoining to \mathbb^, in a suitable manner, a point at infinity. The resulting space is a compact, Hausdorff, and uniquely path-connected topological space which contains \mathbb^(k) as a dense subspace.


References

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External links

* {{nlab, id=Berkovich+space, title=Berkovich space
Institut de Mathématiques de Jussieu Summer School «Berkovich spaces» 2010
Algebraic geometry