 |
 |
 |
 |
|
|
|
|
Course: Math 996, Topics in Commutative Algebra
Time and Place: MWF 11:00 - 11:50pm, Strong Hall 337
Instructor: Professor D. Katz
Office Hours: Snow 501, MWF noon-1:00pm, and via Zoom by appointment
E-mail: dlk53 AT ku dot edu
Prerequisites: A one semester course in commutative algebra or consent of the instructor.
Description: This is a course in commutative algebra covering various topics of interest for students who have had at least one prior course in commutative algebra. Since some of the students in the class have only had a one semester course in commutative algebra, while other students will have had considerably more experience with the subject, I will try to find a way to accommodate as many students as possible. My philosophy for the course will be this: Starting with a particular topic, we will first cover the fundamental results any student in commutative algebra should know that typically appear in a standard commutative algebra text (e.g., Matsumura's Commutative Algebra), and then this will be followed by more advanced results in that topic that might appear in research papers, as opposed to textbooks. For example, one needs to know primary decomposition in order to understand symbolic powers of prime ideals. So, after presenting basic material concerning primary decomposition and stability of asymptotic prime divisors, we will begin a preliminary study of symbolic powers of prime ideals. After presenting some initial results we will then discuss more recent results and open problems related with symbolic powers. The following is a tentative list of topics to be covered, with the aforementioned philosophy in mind.
Introductory topics: Krull intersection and Krull height theorems with applications, including the definition and basic properties of regular local rings
Hilbert rings and Hilbert's Nullstellensatz (as needed for the Zariski-Nagata theorem)
Primary decomposition, asymptotic prime divisors and symbolic powers, including the Eisenbud-Hochster Nullstellensatz with nilpotents theoretm and the Zariski-Nagata theorem, uniform symbolic powers and other recent results and open problems
Cohen-Macaulay rings, basics of local cohomology and applications
Gorenstein rings and applications
Hilbert polynomials, their coefficients and a problem of Vasconcelos
Other topics as time permits, e.g.,syzygies and basic elements
Standard references include: Nagata's Local Rings; the two texts by Matsumura, Commutative Algebra and Commutative Ring Theory; Cohen-Macaulay Rings by Bruns and Herzog; and Commutative Algebra with a View Towards Algebraic geometry by David Eisenbud.
Format: Most class periods will consist of lectures over the material outlined above. Occasionally, we will devote class time to group work on interesting problems related either directly or indirectly to recent topics covered in class. All class materials will be posted to both our course webpage and our Canvas page.
Daily Update : Daily Update
Grading: Later in the semester we will form groups of two or three students from among the members in the class. Each group will read a research paper related to one of the topics we cover in class and give a 30-45 minute presentation of that paper. Each group will turn in a written expository account (10-15 pages in length) of the paper they have read. Final grades will be based upon both the oral presentation and the corresponding write up.
Students with disability
The KU Office of Disability Resources (DR) coordinates accommodations and services for all eligible students with disabilities. If you have a disability and wish to request accommodations and have not contacted DR, please do so as soon as possible. Their office is located in 22 Strong Hall; their phone number is 785-864-2620 (V/TTY). Information about their services can be found at www.disability.ku.edu. Please also contact me privately in regard to your needs in this course.
Policy on religious observances
Any student who has a conflict between the course schedule and a religous holiday should contact the instructor as soon as possible.