4 hours of lectures per week.
Lecturer: Holger Andreas Nielsen
Content:
The solutions of polynomial equations in many variables is a main problem in algebraic geometry and number theory. This leads to the study of commutative rings, their ideals and the modules over such rings. By the process of localization many problems are reduced to local rings.
The first importent invariant is the Krull dimension. A field has Krull dimension 0. A principal ideal domain has Krull dimension 1. A polynomial ring over a field is a unique factorization domain of Krull dimension the number of variables. This is used to give the right measure of size to algebraic curves, surfaces etc.
Generalizations of factorization are most effectively studied by methods applied to finitely generated modules. This leads to class groups and Picard groups. We give a module criterium for a domain to be a unique factorization domain.
The 'nicest' rings are the regular ones. The come up when algebraic curves, surfaces etc. have properties resembling smooth behaviour for differential curves, surfaces etc. A main result states that a local ring is regular if every finitely generated module can be resolved with a finite chain of matrices over the ring. A regular local ring is a unique factorization domain.
The next steps are normal domains coming up in number fields and Cohen-Macaulay rings coming up where algebraic curves have transversal intersection.
Prerequisites: Algebra 2.
Literature: Notes by Birger Iversen.