nLab
analytic spectrum

Definition

For $A$ a ring, its analytic spectrum or Berkovich spectrum ${\mathrm{Spec}}_{\mathrm{an}}A$ 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

$${\mathrm{Spec}}_{\mathrm{an}}A\to {ℝ}_{+}$$Spec_an A \to \mathbb{R}_+

of the form

$$x\mapsto \mid x(a)\mid$$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

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

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

1. $\parallel x\parallel \ge 0$;

2. $\parallel xy\parallel =\parallel x\parallel \parallel y\parallel$;

3. $\parallel x+y\parallel \le \parallel x\parallel +\parallel y\parallel$

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

$${\mid x\mid }_A\le C\parallel x\parallel .$${\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,

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

is the analytic affine line over $k$.

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

Related concepts

• analytic geometry, analytic space

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

• Scott Carnahan, Berkovich spaces I (web)

Lecture notes

Course notes are around definition 4 of

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

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)