In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 case,
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
begins by defining the complex affine space to be
For each
we define
the
ring
Ring 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
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
of
analytic functions 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 derivativ ...
s, i.e. functions on
that can be written as a convergent power series in a
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of each point.
We then define a local model space for
to be
:
with
A
complex analytic space
In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety Complex analytic variety (or just variety) is sometimes required to be irreducible
and (or) reduced or complex analytic space is a general ...
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 geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
.
Berkovich spectrum
A
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
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
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
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 is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exists by inclu ...
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
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
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 that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
of
. The
field of fractions 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
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous ...
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*-al ...
. A point of the Gelfand spectrum is essentially a
homomorphism to
, and its absolute value is the corresponding seminorm in the Berkovich spectrum.
*
Ostrowski's theorem shows that the Berkovich spectrum of the
integers (with the usual norm) consists of the powers
of the usual valuation, for
a
prime or
. If
is a prime then
and if
then
When
these all coincide with the trivial valuation that is
on all non-zero elements. For each
(prime or infinity) we get a branch which is
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
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