2024 Students Enrolled in or before 2015 School of Science Mathematics
Advanced courses in Algebra C
- Academic unit or major
- Mathematics
- Instructor(s)
- Shou Yoshikawa
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Thu
- Class
- -
- Course Code
- ZUA.A333
- Number of credits
- 100
- Course offered
- 2024
- Offered quarter
- 3Q
- Syllabus updated
- Mar 14, 2025
- Language
- English
Syllabus
Course overview and goals
Singularities in positive characteristic is useful for algebraic geometry in all characteristic not only in positive characteristic. The aim of this course together with "Advanced courses in Algebra D" is to introduce the basic notion of Frobenius-regularity with a view towards both classical and modern applications.
Course description and aims
Students are expected to understand the basic notion of Frobenius regularity and quasi-Frobenius-regularity. Looking through concrete examples and applications, students get acquainted with the fundamental importance of singularities in positive characteristic in current research.
Keywords
Commutative ring, Singularities, Frobenius morphisms, Witt 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 | Commutative ring theory in positive characteristic | Details will be provided during each class session |
| Class 2 | Frobenius morphisms and Kunz's theorem | Details will be provided during each class session |
| Class 3 | Frobenius splitting | Details will be provided during each class session |
| Class 4 | Frobenius regularity | Details will be provided during each class session |
| Class 5 | Fedder's criterion | Details will be provided during each class session |
| Class 6 | Test ideal | Details will be provided during each class session |
| Class 7 | Applications for Frobenius regularity | 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.
Matsumura, Hideyuki, Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, 1986.
Karl Schwede, Kevin Tucker, A survey of test ideals, arXiv:1104.2000, 2000.
Evaluation methods and criteria
Course scores are evaluated by homework assignments. Details will be announced during the course.
Related courses
- ZUA.A334 ： Advanced courses in Algebra D
Prerequisites
Basic undergraduate algebra in particular commutative ring theory.
Other
None in particular.