2023 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra B1
- 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-119(H118))
- Class
- -
- Course Code
- MTH.A406
- Number of credits
- 100
- Course offered
- 2023
- Offered quarter
- 2Q
- Syllabus updated
- Jul 8, 2025
- Language
- English
Syllabus
Course overview and goals
The objective of the course is to explain the theory of tight closure, which is a positive-characteristic method introduced by M. Hochster and C. Huneke in 1986. This theory has been developing in connection with the studying of singularities on algebraic varieties and offers powerful tools to solve outstanding problems in commutative ring theory. We will also discuss more recent hot topics in this research, including perfectoid geometry and singularities appearing on local models of Shimura varieties.
Course description and aims
1. Understand basic properties of Frobenius maps
2. Understand the relationship between tight closures and F-singularities
3. Understand the proof and the meaning of Kunz theorem
4. Understand the basics of perfectoid theory
Keywords
Frobenius map, tight closure, F-regular ring, F-rational ring, F-injective ring, perfectoid ring
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 | We will discuss the following topics in the lectures. (1) Frobenius map and Frobenius functor (2) tight closure and colon ideal (3) Kunz theorem (4) F-regular, F-rational, and F-pure rings (5) Frobenius action on local cohomology (6) perfect rings and perfectoid rings (7) big Cohen-Macaulay algebra | Details will be provided during each class session. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 30 minutes preparing for class and another 30 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None required.
Reference books, course materials, etc.
「Cohen-Macaulay Rings」：W.Bruns and J.Herzog
「Foundations of Tight Closure Theory」：M. Hochster
「F-singularities: a commutative algebra appraoch」: L. Ma and T. Polstra
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.A201 ： Introduction to Algebra I
- MTH.A202 ： Introduction to Algebra II
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
Prerequisites
Basic knowledge of some abstract algebra, including rings and modules, is preferable. It is recommended to take "MTH.A401" before taking the current course.