Winter 2002 Ben Richert |
Instructor: Professor B. Richert
Office:. 3852 East Hall
Office phone: 936-4557
Office hours: MWF 1-2, or by appointment.
Email: brichert@umich.edu
Course description: Commutative algebra is the study of commutative rings and their modules. As well as providing the foundation for algebraic geometry, complex analytic geometry, and algebraic number theory, this field has developed into a beautiful and deep theory in its own right, with applications for nearly every algebraist. Algebraic geometers, number theorists, algebraic combinatorialists, lie theorists, and non-commutative algebraists, among others, find it useful. Math 615 is a follow-up on the first course in commutative algebra (614), and our first order of business will be to continue with the topics basic to such a study. These topics will include depth, Cohen-Macaulay rings, Tor and Ext (briefly), Koszul complexes, injective resolutions, regular rings, Gorenstein rings, excellent rings, the structure of complete local rings, etale maps, and possibly Henselian rings. If time permits, we will also study a special topic in the latter portion of the semester. The description must necessarily be vague, but I hope to consider the theory of Hilbert functions (more carefully than in 614). This should include Macaulay's theorem (on maximal Hilbert function growth), Gotzmann's persistence theorem (relating maximal growth to depth), and explore something of the computational flavor of the theory. I also hope to present several open problems which concern questions about Hilbert functions and more generally, about free resolutions.
Prerequisites: Math 614 or consent of instructor.
Course home page: www.math.lsa.umich.edu/~brichert/615.html
Text: Cohen-Macaulay Rings, Revised Edition, Bruns and Herzog.
Coursework: Occasional homework assignments (perhaps up to 4).
Grades: Grades will be based on the homework. Performance in class may be taken into account.
More information may be available on the course home page from time to time.