nLab analytic spectrum

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 $A$ 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 $A \to \mathbb{R}$, the analytic spectrum instead takes points to be multiplicative seminorms ${\vert -\vert} \colon A \to \mathbb{R}_{\geq 0}$ bounded by the norm on the given field.

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

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

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

Definition

For $A$ a normed 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 $x \in A$

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

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

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.

Related concepts

• spectrum (geometry)

• analytic geometry, analytic space

• p-adic geometry

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