Berkovich straight

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In mathematics , the Berkovich straight line is a version of the affine straight line introduced by Vladimir Berkovich , which is particularly useful in p-adic geometry.

motivation

If one defines analytical functions on the p-adic straight line or, more generally, on -manifolds, as those which can be represented locally by convergent power series , then all functions are analytic, because they are totally disconnected . In order to be able to define a meaningful concept of analytical functions, points are added that make the space coherent .

Definition of the Berkovich line

Be a whole body .

The points of are the multiplicative semi-norms on the polynomial ring that continue the absolute amount on . The topology of is the weakest topology with which the mapping for all functions becomes continuous.

Examples

For is , because all multiplicative semi-norms are given by for a .

For an algebraically closed, complete, non-Archimedean field, multiplicative semi-norms are either of the form

for one or

for a . Here referred to .

Berkovich's Classification Theorem

Each corresponds to a descending sequence of nested enclosed spheres . With you get the following classification into four types:

  • Type I: for a
  • Type II: for one
  • Type III: for one
  • Type IV:

properties

The Berkovich straight line is a locally compact Hausdorff space . The type I points corresponding to the points in are close to . The Berkovich line is clearly path-connected; That is, every two points can be connected by an unambiguous shortest path. Type II points are branch points.

literature

  • V. Berkovich: Spectral theory and analytic geometry over non-Archimedean fields , American Mathematical Society, Mathematical Surveys and Monographs 33, 1990

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