Current Topics Seminar

Speaker Lindsay Erickson
Title Introduction to Schemes
Date December 1st, 4:00pm PDL C-36

Schemes are now basic objects in algebraic geometry, and they arose to generalize the older notion of an algebraic variety. If A is a commutative ring with identity, then there is an associated topological space (whose points are the prime ideals of A) called Spec A. Each open subset of Spec A has a collection of algebraic functions on it, just like each open subset of the complex plane has a collection of analytic functions on it. A scheme is formed by carefully sticking together a bunch of spaces of the form Spec A. My goal in this talk is to give you an inkling of the sort of things you'll work with (at least in the beginning) if you choose to study algebraic geometry. I will only assume you have basic familiarity with commutative rings, prime ideals, and topological spaces. (You should know what they are and have exmamples in your head, but it's OK if you just recently learned about them.) This talk will not be rigorous, since it takes much longer than 50 minutes to understand schemes. But I want you to leave with a general implression of what a scheme is, and I hope you'll be able to tell whether or not you find them compelling. This talk is directed at first-year students who haven't chosen a subject area but find algebraic geometry potentially interesting, and also at older students who just want to know what on earth their algebraic geometry friends are talking about. If you've always wanted to hear what a scheme is, this is your chance!