analytic spectrum



For AA a 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

xx(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} \,.


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.



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.


An exposition of examples of Berkovich spectra is in

Lecture notes

Course notes are around definition 4 of

and section 1 of

  • Vladimir Berkovich, 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 November 21, 2013 11:54:19 by Urs Schreiber (