Commutative Algebra

The contents of the course may vary, but it will have a kernel consisting of ideals, modules, chain conditions, the spectrum of a ring, Hilberts Nullestellensatz, associated primes and decomposition theorems, integral elements and rings, valuation rings, Dedekind rings, graded rings, dimension theory. It can also include regular sequences, Koszul complex, and finally regular, Cohen-Macaulay and Gorenstein rings.

1. Knowledge. The contents of the course may vary, but it will have a core consisting of the following: Ideals, prime and maximal ideals, Hilbert's Nullstellensatz, modules, operations on modules, localizations with respect to multiplicatively closed sets and with respect to prime ideals, local rings and local properties, localization of modules, chain conditions, noetherian and artinian rings, dimension theory. It may moreover include Dedekind domains, discrete valuation rings, completions, regular sequences, Koszul complexes, Cohen-Macaulay rings, Gorenstein rings.

2. Skills. The students should learn the topics mentioned above and be able to conduct researches in commutative algebra and its applications.

3. Competence. The students should be able to participate in scientific discussions and begin with own reseach in commutative algebra.

Lectures, possibly as a guided self-study.

The course will be taught as needed. If there are few PhD students, the course is only given as a guided self-study.