Berkovich straight

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 . ${\ displaystyle Q_ {p} ^ {1}}$ ${\ displaystyle Q_ {p}}$ ${\ displaystyle Q_ {p}}$

Definition of the Berkovich line

Be a whole body . ${\ displaystyle K}$

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. ${\ displaystyle A ^ {1, Berk}}$ ${\ displaystyle K \ left [T \ right]}$ ${\ displaystyle K}$ ${\ displaystyle A ^ {1, Berk}}$ ${\ displaystyle | \ cdot | \ mapsto | f |}$ ${\ displaystyle f \ in K \ left [T \ right]}$

Examples

For is , because all multiplicative semi-norms are given by for a . ${\ displaystyle K = \ mathbb {C}}$ ${\ displaystyle A ^ {1, Berk} = \ mathbb {C}}$ ${\ displaystyle f \ mapsto | f (z) |}$ ${\ displaystyle z \ in \ mathbb {C}}$

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

{\ displaystyle f \ mapsto | f (x) |}

for one or ${\ displaystyle x \ in K}$

{\ displaystyle f \ to \ sup _ {x \ in D_ {r} (a)} | f (x) |}

for a . Here referred to . ${\ displaystyle a \ in K, r \ in \ mathbb {R}}$ ${\ displaystyle D_ {r} (a) = \ left \ {x \ in K \ colon \ | xa \ | \ leq r \ right \}}$

Berkovich's Classification Theorem

Each corresponds to a descending sequence of nested enclosed spheres . With you get the following classification into four types: ${\ displaystyle x \ in A ^ {1, Berk}}$ ${\ displaystyle D_ {r_ {i}} (a_ {i})}$ ${\ displaystyle D: = \ bigcap _ {i} D_ {r_ {i}} (a_ {i})}$

Type I: for a ${\ displaystyle D = D_ {0} (a)}$ ${\ displaystyle a \ in K}$
Type II: for one ${\ displaystyle D = D_ {r} (a)}$ ${\ displaystyle a \ in K, r \ in | K ^ {*} |}$
Type III: for one ${\ displaystyle D = D_ {r} (a)}$ ${\ displaystyle a \ in K, r \ not \ in | K ^ {*} |}$
Type IV: ${\ displaystyle D = \ emptyset}$

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. ${\ displaystyle A ^ {1, Berk}}$ ${\ displaystyle K}$ ${\ displaystyle A ^ {1, Berk}}$

literature

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