`PI MU EPSILON PRESENTS`

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**Free resolutions and Hilbert's Syzygy Theorem****
Dr. David Jorgensen
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**Friday, November 21**

**12:00 pm**

**Room 308 Pickard Hall**

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__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.