PI MU EPSILON PRESENTS

Free resolutions and Hilbert's Syzygy Theorem

Dr. David Jorgensen

Friday, November 21

12:00 pm

Room 308 Pickard Hall

Abstract: A commutative ring is like a field except that not all nonzero elements in the ring necessarily have a multiplicative inverse. For example, the set of integers is a commutative ring; 2 does not have a multiplicative inverse in the set of integers. The algebraic structures defined over rings, which are studied ---analogous to vector spaces over a field ---, are called modules. As one might expect, the theory of modules over commutative rings is much more complicated than the theory of vector spaces over a field. The main difference is the following: a minimal set of generators for a vector space over a field has the property of linear independence (such a set is a basis for the vector space). However, a minimal generating set for a module over a ring is most often not linearly independent. In this talk we will give some history on what has been done to study this lack of linear independence for minimal generating sets for modules defined over polynomial rings, the culmination being Hilbert's famous theorem on syzygies. We will also discuss briefly the geometric interpretation of modules over polynomial rings.