Birkhoff’s Theorem from a geometric perspective: A simple example

From Hilbert’s theorem of zeroes, and from Noether’s ideal theory, Birkhoff [1] derived certain algebraic concepts (as explained by Tholen [10]) that have a dual significance in general toposes, similar to their role in the original examples of algebraic geometry. I will describe a simple example that illustrates some of the aspects of this relationship. The dualization from algebra to geometry in the basic Grothendieck spirit can be accomplished (without intervention of topological spaces) by the following method, known as Isbell conjugacy [3], [5].