Nickolas Rollick, Department of Pure Mathematics, University of Waterloo
“The grand scheme of things”
Now that we have been formally introduced to the idea of an affine scheme (at least on the set level), we will start to deepen the acquaintance. In particular, we discuss two important ideas that every self-respecting algebraic geometer knows better than the bottom of their coffee mug. First, we discuss two formulations of Hilbert’s Nullstellensatz and what the theorem says about the “points” of affine n-space over an algebraically closed field k. Secondly, we prove that for any two rings A and B, a ring homomorphism Φ : B → A induces a map from Spec A to Spec B, giving us a contravariant functor from the category of commutative rings to the category of sets. In particular, we take a closer look at this phenomenon in the context of taking quotients and localizations of rings. With any luck, we should also have time to take up the issue of nilpotents in rings, and how they inject some startlingly non-intuitive behavior into the theory of affine schemes.
MC 5413