Instructors: Sasha Voronov (Commutative Algebra, Fall Term) and Bernard Badzioch (Homological Algebra, Spring Term)
Schedule: MWF 9:05 - 9:55 VinH 313
Prerequisite: Math 8201-02 General Algebra or some previous experience with groups, rings, and fields
Textbook: David Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. xvi+785 pp. $69.50; $37.50 paperbound.
Description: Commutative algebra stands at the crossroads of algebra, number theory, and algebraic geometry. It is subsumed by algebraic geometry as the local study of algebraic varieties, somewhat similar to analysis in R^n succumbing to the theory of manifolds. Homological algebra is a powerful algebraic tool used in many fields of mathematics, including commutative and noncommutative algebra, group theory, Lie theory, several complex variables, geometry and topology, PDE, combinatorics, functional analysis, numerical analysis, and mathematical physics, to name a few.
In this one-year course, we will study commutative rings and modules over them, Noetherian rings, Krull dimension theory, Noether normalization, the so-called Nullstellensatz, the spectrum of a ring, rings of fractions and localization, primary decomposition, discrete valuation rings, normal integral domains, regular rings and Cohen-Macaulay rings. The geometric view of a commutative ring as the ring of functions on a space will be emphasized.
The homological algebra part of the course will cover complexes, homology, resolutions and derived functors. These notions will be put into the context of two different axiomatic approaches to homological algebra: via triangulated categories and via closed model categories. Additional topics will include Koszul complex, Hochschild homology and cyclic homology. Applications to commutative algebra, algebraic geometry and topology will be discussed.
Requirements: There will be homework, but no exams. One in-class topic presentation per year will be expected.
More information: Contact the instructors at
voronov@math.umn.edu
VinH 324
624-0355
badzioch@math.umn.edu
VinH 302
625-9817