2026 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra A
- 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-101(H116))
- Class
- -
- Course Code
- MTH.A401
- Number of credits
- 100
- Course offered
- 2026
- Offered quarter
- 1Q
- Syllabus updated
- Mar 5, 2026
- Language
- English
Syllabus
Course overview and goals
This course provides an introduction to the fundamentals of perfectoid theory, a field of growing importance across algebraic geometry, number theory, and commutative algebra. In conjunction with the curriculum of "Advanced Topics in Algebra B," the lectures will cover the essentials of commutative algebra, Banach modules, and adic spaces.
Course description and aims
This course explores the structural properties of commutative rings across different characteristics. We begin by utilizing the Frobenius endomorphism as a fundamental tool for analyzing rings of positive characteristic. Building on this, we introduce the theory of Tight Closure, developed by Hochster and Huneke, to study singularities and homological conjectures.The final portion of the course bridges the gap between positive characteristic and characteristic zero by examining mixed characteristic rings. We will delve into the foundations of Perfectoid Theory, focusing on how non-Noetherian commutative algebra facilitates the "Almost Mathematics" necessary to relate these disparate algebraic worlds.
Keywords
Positive characteristic、Frobenius map、Cohen-Macaulay ring、perfect closure
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: Basics on commutative rings |
Details will be provided during each class sessions |
| Class 2 | Frobenius map and singularities１ |
Details will be provided during each class sessions |
| Class 3 | Frobenius map and singularities２ |
Details will be provided during each class sessions |
| Class 4 | Cohen-Macaulay ring |
Details will be provided during each class sessions |
| Class 5 | Big Cohen-Macaulay algebra |
Details will be provided during each class sessions |
| Class 6 | Perfect closure, algebra modification |
Details will be provided during each class sessions |
| Class 7 | Construction of big Cohen-Macaulay algebra |
Details will be provided during each class sessions |
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.
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.A402 ： Advanced topics in Algebra B
- ZUA.A331 ： Advanced courses in Algebra A
- ZUA.A332 ： Advanced courses in Algebra B
Prerequisites
Basic undergraduate algebra, commutative ring and modules
Other
None in particular.