Non-Archimedean Banach Algebras

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Non-Archimedean Banach Algebras

Semester
SoSe 2025

Lecturer
Klaus Künnemann

Type of course (Veranstaltungsart)
Vorlesung

German title
Non-Archimedean Banach Algebras

Contents
This course gives a first introduction to non-archimedean Banach algebras and could have also be
named Algebraic Number Theory II. It requires a good knowledge of algebra and commutative algebra
but basically no prerequisites from Algebraic Number Theory I. Non-archimedean Banach algebras
play a crucial role in modern Non-Archimedean Analytic Geometry and in Scholze's theory of
Perfectoid Spaces. We are mainly interested in affinoid algebras which can be described as quotients
of algebras of convergent power series. We follow the approach by Berkovich developed in his book
'Spectral theory and analytic geometry over non-archimedean fields' and subsequent papers. We will
discuss Banach spaces over non-archimedean fields, spectra of commutative Banach rings, affinoid
algebras and their algebraic properties, reduction, boundary and interior. This course serves as
an introduction to my course 'Non-Archimedean Analytic Geometry' in the coming winter term.

Recommended previous knowledge
Algebra and Commutative Algebra

Time/Date
Monday, Thursday 10h15-12h00

Location
Monday M103, Thursday M101

Registration

Preliminary registration for the organisation of exercise classes: at the end of the previous
semester via EXA or LSF (see announcement by the department)
Registration for the exercise classes: You can register via GRIPS for the exercise classes in
the first week of the teaching period.
Registration for course work/examination/ECTS: FlexNow

Course work (Studienleistungen)

Successful participation in the exercise classes: Active participation and the presentation of
solutions to exercises: Each participant has to present at least two solutions, at least one
from the exercise sheets 1-6 and at least one from the exercise sheets 7-12. Furthermore
written solutions to the exercises have to be submitted (at least 30% successful solutions from
sheet 1-6 and at least 30% successful solutions from sheet 7-12).

Examination (Prüfungsleistungen)

Oral exam: Duration: 30 min, Date: by individual appointment , re-exam: Date: by individual
appointment

Modules
BV, MV, MArGeo

ECTS
9 LP