When we have an algebraic variety we can identifie the points of the variety with maximal ideals of the coordinate ring.
I would like to know why is more natural to define the main structure of the theory of schemes: the affine scheme, with the prime ideals and not with the maximal ones.
When he was creating the theory of schemes why did he decide to work with the normal spectrum instead of the maximal one?
(As you can see I dont have an strong background of Algebraic Geometry, I just want to have some intuition)
In which sense the schemes generalize the notion of variety and why is better to work with this notion?