Higher Algebra
A.Y. 2024/2025
Learning objectives
The course consists in an introduction to non-archimedean arithmetic geometry using the language of adic spaces, formal schemes and condensed rings. We will provide some fundamental examples such as Tate's analytic varieties, Scholze's perfectoid spaces and the Fargues-Fontaine curve. We will discuss applications of this formalism to the study of p-adic cohomology theories.
Expected learning outcomes
By the end of the course, students will be able to handle the language and techniques of modern non-archimedean geometry, which are the foundations of the recent developments in p-adic Hodge theory.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
Introduction to non-archimedean analytic geometry and its applications to arithmetic geometry.
- Basic definitions of the theory of adic spaces: valuations, Huber rings. Examples: perfectoid spaces, tilting, the Fargues-Fontaine curve.
- Basic definitinons of Raybaud's theory via formal schemes and blow-ups, specialization maps.
- de Rham cohomology for rigid varieties: dagger structures, analytic spaces à la Clausen-Scholze, homotopy theory.
- Rigid cohomology.
- Basic definitions of the theory of adic spaces: valuations, Huber rings. Examples: perfectoid spaces, tilting, the Fargues-Fontaine curve.
- Basic definitinons of Raybaud's theory via formal schemes and blow-ups, specialization maps.
- de Rham cohomology for rigid varieties: dagger structures, analytic spaces à la Clausen-Scholze, homotopy theory.
- Rigid cohomology.
Prerequisites for admission
Basics of commutaive algebra and algebraic geometry (included in the courses Algebra 3-4 and Projective Algebraic Geometry/Scheme Theory).
Teaching methods
Lectures.
Teaching Resources
Roland Huber "Continuous Valuations" & "A Generalization of Formal Schemes and Rigid Analytic Varieties"
Sophie Morel "Adic Spaces" Lecture Notes
Matthew Morrow "Adic and perfectoid spaces" Lecture Notes
Bhargav Bhatt "Lecture notes for a class on perfectoid spaces"
Peter Scholze "Perfectoid Spaces"
Jean-Marc Fontaine "Perfectoides, Presque-Purete et Monodromie-Poids" (Bourbaki)
Torsten Wedhorn "Adic Spaces" Lecture Notes
Peter Scholze, Jared Weinstein "Berkeley Lectures on p-adic Geometry"
Kazuhiro Fujiwara, Fumiharu Kato "Foundations of Rigid Geometry I"
Peter Scholze "Lectures on Analytic Geometry"
Sophie Morel "Adic Spaces" Lecture Notes
Matthew Morrow "Adic and perfectoid spaces" Lecture Notes
Bhargav Bhatt "Lecture notes for a class on perfectoid spaces"
Peter Scholze "Perfectoid Spaces"
Jean-Marc Fontaine "Perfectoides, Presque-Purete et Monodromie-Poids" (Bourbaki)
Torsten Wedhorn "Adic Spaces" Lecture Notes
Peter Scholze, Jared Weinstein "Berkeley Lectures on p-adic Geometry"
Kazuhiro Fujiwara, Fumiharu Kato "Foundations of Rigid Geometry I"
Peter Scholze "Lectures on Analytic Geometry"
Assessment methods and Criteria
Seminar on an assigned topic.
MAT/02 - ALGEBRA - University credits: 6
Lessons: 42 hours
Professor:
Vezzani Alberto
Shifts:
Turno
Professor:
Vezzani AlbertoProfessor(s)