Berkovich space explained
In 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 begins by defining the complex affine space to be
For each
we define
the
ring of
analytic functions on
to be the ring of
holomorphic functions, i.e. functions on
that can be written as a convergent
power series in a
neighborhood of each point.
We then define a local model space for
to be
X:=\{x\inU:f1(x)= … =fn(x)=0\}
with
l{O}X=l{O}U/(f1,\ldots,fn).
A
complex analytic space is a locally ringed
-space
which is locally isomorphic to a local model space.
When
is a
complete non-Archimedean field, we have that
is
totally disconnected. 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.
Berkovich spectrum
A seminorm on a ring
is a non-constant function
such that
\begin{align}
|0|&=0\\
|1|&=1\\
|f+g|&\leqslant|f|+|g|\\
|fg|&\leqslant|f||g|
\end{align}
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 of multiplicative seminorms on
that are bounded by the norm of
.
The Berkovich spectrum is equipped with the weakest topology such that for any
the map
\begin{cases}\varphif:l{M}(A)\to\R\ | ⋅ |\mapsto|f|\end{cases}
is continuous.
The Berkovich spectrum of a normed ring
is
non-empty if
is
non-zero and is
compact if
is complete.
If
is a point of the spectrum of
then the elements
with
form a
prime ideal 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
\rho(f)=\limn\toinfty\left\|fn\right
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. A point of the Gelfand spectrum is essentially a
homomorphism to
, and its absolute value is the corresponding seminorm in the Berkovich spectrum.
of the usual valuation, for
a
prime or
. If
is a prime then
0\leqslant\varepsilon\leqslantinfty,
and if
then
0\leqslant\varepsilon\leqslant1.
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 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
extending the norm on
.
The Berkovich affine space is equipped with the weakest topology such that for any
the map
taking
to
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
as a map
with
, which is a local limit of rational functions, i.e., such that every point
has an open neighborhood
with the following property:
\forall\varepsilon>0\existg,h\inl{k}[x1,\ldots,xn]: \forallx'\inU'\left(h(x') ≠ 0 \land \left|f(x')-
\right|<\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
In the case where
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
is an
algebraically closed non-Archimedean field, complete with respects to its valuation, one can describe all the points of the affine line.
.
The space
is a locally compact,
Hausdorff, and uniquely path-connected topological space which contains
as a
dense subspace.
One can also define the Berkovich projective line
by adjoining to
, in a suitable manner, a point at infinity. The resulting space is a compact, Hausdorff, and uniquely path-connected topological space which contains
as a dense subspace.
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