Commutative Algebra

Objectives:

The course develops the theory of commutative rings. These rings are of fundamental significance since geometric and number theoretic ideas is described algebraically by commutative rings.

Content:

One studies ideals in commutative rings, chain conditions for ideals, localization of commutative rings, modules over commutative rings and numerical invariants of commutative rings and modules. Important results concern tensor products and exact sequences of module, noetherian rings and Hilbert basis theorem, Nullstellensatz, Noether normalization, and primary decomposition of ideals. One develops the theory of Gröbner bases, Hilbert series and Hilbert polynomials, and dimension theory for local rings.

On completion of the course the student should have the following learning outcomes defined in terms of knowledge, skills and general competence:

Knowledge

The student

• Knows basic definitions concerning elements in rings, classes of rings, and ideals in commutative rings.
• Know constructions like tensor product and localization, and the basic theory for this.
• Know basic theory for noetherian rings and Hilbert basis theorem.
• Know basic theory for integral dependence, and the Noether normalization lemma.
• Have insight in the correspondence between ideals in polynomial rings, and the corresponding geometric objects: affine varieties.
• Know basic theory for support and associated prime ideals of modules, and know primary decomposition of ideals in noetherian rings.
• Know the theory of Gröbner bases and Buchbergers algorithm.
• Know the theory of Hilbert series and Hilbert polynomials.
• Know dimension theory of local rings.

Skills

The student

• Can use algebraic tools which are important for many problems and much theory development in algebra, algebraic geometry, number theory, and topogy.
• Have solid experience and training in reasoning with abstract and general algebraic structures.

General competence

The student

• Has insight in the most important algebraic theory which is used in other parts of mathematics.
• Has insight in the mathematics that is used in computer algebra.
• See the usefulness of abstract theory development so that different parts of mathematics, like number theory and algebraic geometry, can be described in the same framework