2026 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra B
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Kazuma Shimomoto
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Thu (M-102(H115))
- Class
- -
- Course Code
- MTH.A402
- Number of credits
- 100
- Course offered
- 2026
- Offered quarter
- 2Q
- Syllabus updated
- Mar 5, 2026
- Language
- English
Syllabus
Course overview and goals
Building upon the material covered in 'Advanced Topics in Algebra A,' this course introduces the foundations of Almost Mathematics. Furthermore, after discussing continuous valuations, Banach rings、Tate rings, and perfectoid rings, we will provide an exposition of the Almost Purity Theorem.
Course description and aims
Students will become proficient in Almost Mathematics and the handling of perfectoid algebras via the Frobenius map. Additionally, the course aims to deepen understanding of the concept of perfectoid spaces and the tilting equivalence (tilting functor).
Keywords
Almost mathematics、Banach ring、Tate ring、perfectoid space、tilting、almost purity theorem
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Introduction to almost mathematics |
Details will be provided during each class session |
| Class 2 | Banach ring, continuous valuation |
Details will be provided during each class session |
| Class 3 | Adic space, rational localization１ |
Details will be provided during each class session |
| Class 4 | Adic space, rational localization２ |
Details will be provided during each class session |
| Class 5 | Perfectoid space |
Details will be provided during each class session |
| Class 6 | Almost etale extension, tilting correspondence |
Details will be provided during each class session |
| Class 7 | Almost purity theorem |
Details will be provided during each class session |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to explore references provided in lectures and other materials.
Textbook(s)
None required.
Reference books, course materials, etc.
M.Hochster: Foundations of tight closure theory
T.Polstra and L.Ma: F-singularities: A commutative algebra approach(https://www.math.purdue.edu/~ma326/F-singularitiesBook.pdf)
K.Shimomoto: Lectures on perfectoid geometry for commutative algbraists
O.Gabber and L.Ramero: Almost ring theory
Evaluation methods and criteria
Course scores are evaluated by homework assignments. Details will be announced during the course.
Related courses
- MTH.A401 ： Advanced topics in Algebra A
- ZUA.A331 ： Advanced courses in Algebra A
- ZUA.A332 ： Advanced courses in Algebra B
Prerequisites
MTH.A401 ： Advanced topics in Algebra A
Other
None in particular.