Berkovich space

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.

MotivationEdit

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.

Berkovich spectrumEdit

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 quotient field 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

 

ExamplesEdit

  • The spectrum of a field complete with respect to a valuation is a single point corresponding to its valuation.
  • If   is a commutative C*-algebra then the Berkovich spectrum is the same as the Gelfand spectrum. A point of the Gelfand spectrum is essentially a homomorphism to  , and its absolute value is the corresponding seminorm in the Berkovich spectrum.
  • Ostrowski's theorem shows that the Berkovich spectrum of the integers (with the usual norm) consists of the powers   of the usual valuation, for   a prime or  . If   is a prime then   and if   then   When   these all coincide with the trivial valuation that is   on all non-zero elements. For each   (prime or infinity) we get a branch which is homeomorphic to a real interval, the branches meet at the point corresponding to the trivial valuation. The open neighborhoods of the trivial valuations are such that they contain all but finitely many branches, and their intersection with each branch is open.

Berkovich affine spaceEdit

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 spectrums of rings of power series that converge in some ball (so it is locally compact).

We define an analytic function on an open subset   is 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.

Berkovich affine lineEdit

The 1-dimensional Berkovich affine space is called the Berkovich affine line. When   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.

ReferencesEdit

  • Baker, Matthew; Conrad, Brian; Dasgupta, Samit; Kedlaya, Kiran S.; Teitelbaum, Jeremy (2008), Thakur, Dinesh S.; Savitt, David (eds.), p-adic geometry, University Lecture Series, 45, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4468-7, MR 2482343
  • Baker, Matthew; Rumely, Robert (2010), Potential theory and dynamics on the Berkovich projective line, Mathematical Surveys and Monographs, 159, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4924-8, MR 2599526
  • Berkovich, Vladimir G. (1990), Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1534-2, MR 1070709
  • Berkovich, Vladimir G. (1993), "Étale cohomology for non-Archimedean analytic spaces", Publications Mathématiques de l'IHÉS (78): 5–161, ISSN 1618-1913, MR 1259429

External linksEdit