This course is a basic introduction to commutative algebra for students who have completed the alpha sequence in algebra. Topics covered will include localization of rings and modules, primary decomposition of ideals, dimension theory for rings, tensor products and flatness, completion of rings. Topics important in number theory, including Dedekind domains and Hensel's Lemma, will be covered. We will also cover a number of topics essential in algebraic geometry, including the prime spectrum of a ring, Hilbert's nullstellensatz, and Noether normalization.
If you enjoyed Math 593/594, this is a good next course to take.
The course will be seriously graded, with bi-weekly problem sets. There are two ``textbooks:'' Professor Hochster's course notes for 614 and David Eisenbud's text "Commutative algebra with a view towards algebraic geometry" (Springer). A good back-up source is "Introduction to Commutative algebra" by Atiyah and MacDonald (Addison--Wesley).