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
For each
we define
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
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
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
to be
:
with
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
-space
which is locally isomorphic to a local model space.
When
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
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
, and also gives back the usual definition over
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
is a non-constant function
such that
:
for all
. It is called multiplicative if
and is called a norm if
implies
.
If
is a normed ring with norm
then the Berkovich spectrum of
, denoted
, 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
that are bounded by the norm of
.
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
the map
:
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
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
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
is complete.
If
is a point of the spectrum of
then the elements
with
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
. 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
and the image of an element
is denoted by
. The field
is generated by the image of
.
Conversely a bounded map from
to a complete normed field with a multiplicative norm that is generated by the image of
gives a point in the spectrum of
.
The spectral radius of
:
is equal to
:
Examples
* The spectrum of a field complete with respect to a valuation is a single point corresponding to its valuation.
* If
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
, 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:
and
(
a prime). Here
,
is the
-adic norm for which
,
is the seminorm induced by the trivial norm
on
, and
is the trivial norm on
, i.e., the norm which sends all nonzero elements to 1. For each
(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
is a field with a
valuation, then the ''n''-dimensional Berkovich affine space over
, denoted
, is the set of multiplicative seminorms on