Connected Components and Separable Algebras

Abstract

The introduction of Spec A has given us a general definition of connected components, but a more subtle problem remains. Take for example µ₃, represented by A = k[X]/(X³ − 1). Over the reals there are two points in Spec A, reflecting the decomposition X³ − 1 = (X – 1)(X² + X + 1). But over the complex numbers the group is isomorphic to ℤ/3ℤ, and we get three components. Thus base extension can create additional idempotents. To have a complete theory of connected components, we need a fancier version that will detect these “potential idempotents.” Over fields—and for the rest of this part we assume k is a field—the question can be handled using separable algebras.