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In mathematics, a Berkovich space, introduced by Berkovich (1990), 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.
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.
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
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.
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.
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