Author: Eiko
Tags: p-adic geometry, Berkovich space
Time: 2024-12-17 14:19:59 - 2024-12-17 14:19:59 (UTC)
References:
Spectral Theory and Analytic Geometry over Non-Archimedean Fields by Vladimir Berkovich
Math 731 Topics in Algebraic Geometry I Berkovich Spaces by Mattias Jonsson
Review
There are various ways to deal with non-archimedean geometry, including
Rigid Analytic Spaces by J.Tate, they use Grothendieck topology.
It is good for category of sheaves, but does not follow a direct intuitive comparison with classical complex field.
Berkovich Spaces, they use the theory of valuations.
They are elegant objects possessing topological properties similar to that of complex analytic spaces and are sufficient for homology theory in the usual sense.
Adic Spaces, they use the theory of Huber rings.
Berkovich Affine Spaces
Similar to the theory of valuations, where the valuations in a field like corresponds to points in , Berkovich spaces generalize this idea,, to use seminorms on a valued field to define a space.
Think of each valuation corresponds to the valuation ring with a maximal ideal.
Seminorm
A norm on a field is a function satisfying the following properties:
A seminorm however, does not require the regularity property, i.e. you can have with .
Berkovich Affine Space Is The Set of Seminorms
Let be a complete valued field. The Berkovich affine space is defined as the set of multiplicative seminorms on extending the norm on .
The topology on is the weakest topology making the evaluation maps continuous for all .
is Hausdorff, locally compact, and path-connected.
, because for any point , the evaluation at followed by the norm on gives us a seminorm!
and you see that it is a seminorm because you cannot reasonably require to imply .
If we take with the usual absolute value, then the Berkovich affine space is just the usual complex affine space.
Berkovich Affine Line
For simplicity assume is algebraically closed, this simplifies valuations on to evaluating norms on the linear factors. Simplify further we elt be the trivial norm. For be a point in our line, there are some cases:
is the trivial norm on .
All .
, since is trivial, we have for all .
All .
There exists such that , and then for any other since we have .
Only one , all others are .