**
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.

MC 5417