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 formula_1 For each formula_2 we define formula_3 the ring of analytic functions on formula_4 to be the ring of holomorphic functions, i.e. functions on formula_4 that can be written as a convergent power series in a neighborhood of each point.
We then define a local model space for formula_6 to be 
with formula_8 A complex analytic space is a locally ringed formula_9-space formula_10 which is locally isomorphic to a local model space.
When formula_11 is a complete non-Archimedean field, we have that formula_11 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 formula_11, and also gives back the usual definition over formula_14
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 formula_15 is a non-constant function formula_16 such that 
for all formula_18. It is called multiplicative if formula_19 and is called a norm if formula_20 implies formula_21.
If formula_15 is a normed ring with norm formula_23 then the Berkovich spectrum of formula_15, denoted formula_25, is the set of multiplicative seminorms on formula_15 that are bounded by the norm of formula_15. 
The Berkovich spectrum is equipped with the weakest topology such that for any formula_28 the map 
is continuous.
The Berkovich spectrum of a normed ring formula_15 is non-empty if formula_15 is non-zero and is compact if formula_15 is complete.
If formula_33 is a point of the spectrum of formula_15 then the elements formula_35 with formula_36 form a prime ideal of formula_15. The quotient field 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 formula_38 and the image of an element formula_39 is denoted by formula_40. The field formula_38 is generated by the image of formula_15. 
Conversely a bounded map from formula_15 to a complete normed field with a multiplicative norm that is generated by the image of formula_15 gives a point in the spectrum of formula_15.
The spectral radius of formula_46
is equal to 
Berkovich affine space.
If formula_11 is a field with a valuation, then the "n"-dimensional Berkovich affine space over formula_11, denoted formula_63, is the set of multiplicative seminorms on formula_64 extending the norm on formula_11. 
The Berkovich affine space is equipped with the weakest topology such that for any formula_66 the map formula_67 taking formula_68 to formula_69 is continuous.
This is not a Berkovich spectrum, but is an increasing union of the Berkovich spectrums of rings of power series that converge in some ball (so it is locally compact).
We define an analytic function on an open subset formula_70 is a map 
with formula_72 which is a local limit of rational functions, i.e., such that every point formula_73 has an open neighborhood formula_74 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 formula_76
In the case where formula_77 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 formula_11 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 formula_79 .
The space formula_80 is a locally compact, Hausdorff, and uniquely path-connected topological space which contains formula_11 as a dense subspace.
One can also define the Berkovich projective line formula_82 by adjoining to formula_80, in a suitable manner, a point at infinity. The resulting space is a compact, Hausdorff, and uniquely path-connected topological space which contains formula_84 as a dense subspace.