nLab
analytic spectrum

Contents

Idea

The concept of analytic spectrum is a realization of the concept of spectrum (geometry) in the context of non-archimedean analytic geometry. Given an affinoid algebra AA over a non-archimedean field, then the concept of spectrum of a commutative ring whose points are prime ideals/maximal ideals does not produce a sensible space that admits analytic geometry. Rather, instead of regarding points of the spectrum as ring homomorphisms to AA \to \mathbb{R}, the analytic spectrum instead takes points to be multiplicative seminorms ||:A 0{\vert -\vert} \colon A \to \mathbb{R}_{\geq 0} bounded by the norm on the given field.

If Spec an(A)Spec_{an}(A) is the set of all such multiplicative seminorms, then for xSpec an(A)x \in Spec_{an}(A) a point one writes the corresponding seminorm as || x{\vert - \vert}_x and thinks of it as being the norm on the function algebra on Spec(A)Spec(A) which is given by “evaluating functions at xx and then applying the field norm to that”.

One turns this Spec an(A)Spec_{an}(A) into a topological space in the usual way by choosing the weakest topology such that under this assignment the original elements of AA become continuous function on Spec an(A)Spec_{an}(A).

Globalizing this analytic spectrum construction leads to the concept of Berkovich analytic space.

Definition

For AA a normed ring, its analytic spectrum or Berkovich spectrum Spec anASpec_an A is the set of all non-zero multiplicative seminorms on AA, regarded as a topological space when equipped with the weakest topology such that all functions

Spec anA + Spec_{an} A \to \mathbb{R}_+

of the form

x|x(a)| x \mapsto {\vert x(a)\vert}

for aAa \in A are continuous.

If AA is equipped with the structure of a Banach ring, one takes the bounded multiplicative seminorms.

So a point in the analytic spectrum of AA corresponds to a non-zero function

:A {\Vert -\Vert} : A \to \mathbb{R}

to the real numbers, such that for all x,yAx, y \in A

  1. x0{\Vert x \Vert} \geq 0;

  2. xy=xy{\Vert x y \Vert} = {\Vert x \Vert} {\Vert y \Vert};

  3. x+yx+y{\Vert x + y \Vert} \leq {\Vert x \Vert} + {\Vert y \Vert}

and boundedness means that there exists C>0C \gt 0 such that for all xAx \in A

|x| ACx. {\vert x \vert}_A \leq C {\Vert x\Vert} \,.

(e.g. Berkovich 09, def. 1.2.3)

Examples

Affine line

For kk a field and k[T]k[T] the polynomial ring over kk in one generator,

𝔸 k:=Spec ank[T] \mathbb{A}_k := Spec_{an} k[T]

is the analytic affine line over kk.

If k=k = \mathbb{C}, then 𝔸 k=\mathbb{A}_k = \mathbb{C} is the ordinary complex plane.

References

Original

The notion originates in

  • Vladimir Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, (1990) 169 pp.

Expositions and Lecture notes

Introductory exposition of the Berkovich analytic spectrum includes

  • Sarah Brodsky, Non-archimedean geometry, brief lecture notes, 2012 (pdf)

  • Scott Carnahan, Berkovich spaces I (web)

  • Jérôme Poineau, Global analytic geometry, pages 20-23 in EMS newsletter September 2007 (pdf)

  • Frédéric Paugam, section 2.1.4 of Global analytic geometry and the functional equation (2010) (pdf)

  • Vladimir Berkovich, section 1 of Non-archimedean analytic spaces, lectures at the Advanced School on pp-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)

Revised on July 18, 2014 00:25:49 by Urs Schreiber (82.136.246.44)