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 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 , the analytic spectrum instead takes points to be multiplicative seminorms bounded by the norm on the given field.
If is the set of all such multiplicative seminorms, then for a point one writes the corresponding seminorm as and thinks of it as being the norm on the function algebra on which is given by “evaluating functions at and then applying the field norm to that”.
One turns this into a topological space in the usual way by choosing the weakest topology such that under this assignment the original elements of become continuous function on .
Globalizing this analytic spectrum construction leads to the concept of Berkovich analytic space.
For a normed ring, its analytic spectrum or Berkovich spectrum is the set of all non-zero multiplicative seminorms on , regarded as a topological space when equipped with the weakest topology such that all functions
of the form
for are continuous.
If is equipped with the structure of a Banach ring, one takes the bounded multiplicative seminorms.
So a point in the analytic spectrum of corresponds to a non-zero function
to the real numbers, such that for all
and boundedness means that there exists such that for all
(e.g. Berkovich 09, def. 1.2.3)
For a field and the polynomial ring over in one generator,
is the analytic affine line over .
If , then 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.
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 -adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)