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When we have an algebraic variety we can identify the points of the variety with maximal ideals of the coordinate ring.

I would like to know why it is more natural to define the main structure of the theory of schemes, the affine scheme, with prime ideals and not with maximal ones.

When Grothendieck was creating 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?

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Dear Sophie: this is a long story, very well told here. It has to do with the inverse image of primes under ring morphisms being primes, with nilpotent elements, with getting rid of base fields for arithmetical applications and many other wonders. Also, let me emphasize that they did not create scheme theory, He did (Alexander Grothendieck). –  Georges Elencwajg Apr 17 '12 at 8:57
Dear Sophie: I don't know (by far!) the subject as well as Georges, but I think the best one sentence answer would be something like: "you want Spec to be a functor". (No claim of originality, needless to say.) –  Pierre-Yves Gaillard Apr 17 '12 at 9:13
Yes, if you have a function between rings you cannot define a natural function between the spectrum of maximal ideals of them. That's why we have to see the varieties as schemes over a field . Thanks! –  André Apr 17 '12 at 9:24

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