# Contents

## Definition

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

$Spec_an A \to \mathbb{R}_+$

of the form

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

for $a \in A$ are continuous.

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

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

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

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

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

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

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

and boundedness means that there exists $C \gt 0$ such that for all $x \in A$

${\vert x \vert}_A \leq C {\Vert x\Vert} \,.$

## Examples

### Affine line

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

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

is the analytic affine line over $k$.

If $k = \mathbb{C}$, then $\mathbb{A}_k = \mathbb{C}$ 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

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 $p$-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)

Revised on November 21, 2013 11:54:19 by Urs Schreiber (188.200.54.65)