Berkovich space
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

with  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

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

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 

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.
 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 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:

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.

There is a canonical embedding  .

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.

References

External links
 
 Institut de Mathématiques de Jussieu Summer School «Berkovich spaces» 2010

Algebraic geometry