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.
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.
Commutative ring, Singularities, Frobenius morphisms, Witt ring.
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course
Course schedule | Required learning | |
---|---|---|
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 |
To enhance effective learning, students are encouraged to explore references provided in lectures and other materials.
None required.
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.
Course scores are evaluated by homework assignments. Details will be announced during the course.
Basic undergraduate algebra in particular commutative ring theory.
None in particular.