Winter Term 2012/2013 - Commutative Algebra

Link to the moodle class room.

There you find more information about the lecture, the lecture notes and the problem sheets.

Targets

In number theory one is naturally led to study more general numbers than just the classical integers and, thus, to introduce the concept of integral elements in number fields. The rings of integers in number fields have certain very beautiful properties (such as the unique factorisation of ideals) which characterise them as Dedekind rings. Parallely, in geometry one studies affine varieties through their coordinate rings. It turns out that the coordinate ring of a curve is a Dedekind ring if and only if the curve is non-singular (e.g. has no self intersection). With this in mind, we shall work towards the concept and the characterisation of Dedekind rings. Along the way, we shall introduce and demonstrate through examples basic concepts of algebraic geometry and algebraic number theory. Moreover, we shall be naturally led to treat many concepts from commutative algebra.

Content

Depending on the previous knowledge of the audience, the lecture will cover all or parts of the following topics:

• General concepts in the theory of commutative rings:
  • rings, ideals and modules,
  • Noetherian rings,
  • tensor products,
  • localisation,
  • completion,
  • dimension.
• Number rings:
  • integral extensions,
  • ideals and discriminants,
  • Noether's normalisation theorem,
  • Dedekind rings,
  • unique ideal factorisation.
• Plane Curves:
  • affine space,
  • coordinate rings and Zariski topology,
  • Hilbert's Nullstellensatz,
  • resultant and intersection of curves,
  • morphisms of curves,
  • singular points.

Literature

There are many books about the content of the lecture, for example:

• E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry.
• M. F. Atiyah, I. G. Macdonald. Introduction to Commutative Algebra, Addison-Wesley Publishing Company.
• Dino Lorenzini. An Invitation to Arithmetic Geometry, Graduate Studies in Mathematics, Volume 9, American Mathematical Society.

Last modification: 17 September 2012.