Student Algebra seminar

Robert Garbary, Pure Mathematics Department, University of Waterloo

"The Zariski Topology"

For a ring R, let Spec(R) denote the set of its prime ideals. We are going to topologize Spec(R) by demanding that R is the ring of continuous functions (into where?) defined on Spec(R). This is called the Zariski Topology. It is so named because, in the case that R is the coordinate ring of an affine variety X, we will see that Spec(R) is almost equal to (whatever that means) X, and that the topology on Spec(R) is almost the same as the usual Zariski topology on X. This is the first step in saying what a scheme is.