Birkhoff's theorem from a geometric perspective: A simple example

From Hilbert's theorem of zeroes‎, ‎and from Noether's ideal theory‎, ‎Birkhoff derived certain algebraic concepts (as explained by Tholen) 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 the intervention of topological spaces) by the following method‎, ‎known as Isbell conjugacy.