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
the ring of analytic 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
A complex analytic space is a locally ringed
is a complete non-Archimedean field, we have that
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.
, is the set of multiplicative seminorms on
The Berkovich spectrum is equipped with the weakest topology such that for any
The Berkovich spectrum of a normed ring
Conversely a bounded map from
gives a point in the spectrum of
is a field with a valuation, then the n-dimensional Berkovich affine space over
The Berkovich affine space is equipped with the weakest topology such that for any
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
, which is a local limit of rational functions, i.e., such that every point
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
this will give the same objects as described in the motivation section.
The 1-dimensional Berkovich affine space is called the Berkovich affine line.
is an algebraically closed non-Archimedean field, complete with respects to its valuation, one can describe all the points of the affine line.
is a locally compact, Hausdorff, and uniquely path-connected topological space which contains
One can also define the Berkovich projective line
, in a suitable manner, a point at infinity.
The resulting space is a compact, Hausdorff, and uniquely path-connected topological space which contains