David Urbanik, Department of Pure Mathematics, University of Waterloo
"Chevalley's Theorem on Constructible Sets in the Language of Schemes"
We continue working through Vakil's notes, this time covering section 7.4 on Chevalley's Theorem on Constructible Sets. Chevalley's theorem gives a sensible answer to the question: given a sufficiently "nice" scheme X and a correspondingly "nice" morphism X to Y, what can we say about the image of X in Y? The answer is that the image is a so-called "constructible set", which we will see is a natural notion that generalizes open and closed sets. Chevalley's theorem will also give us a proof of the Nullstellensatz, and help us develop intuition for the types of scheme morphisms we have defined so far.