I just started looking at the notes https://www.jmilne.org/math/CourseNotes/iAG200.pdf. And in the Appendix where they review some algebraic geometry they define
sets of the form
$$
Z(\mathfrak{a}) = \{\mathfrak{m} : \mathfrak{a} \subseteq \mathfrak{m} \}
$$
where $\mathfrak{m}$ is a maximal ideal of $A$, a finitely generated $k$-algebra ($k$ is a field) as the closed sets of the Zariski topology.
I am used to seeing the Zariski topology defined in terms of prime ideals, and not maximal ideals like this.
I was wondering if someone could explain me why this makes more sense? or what are some of the differences I should keep in mind?
Thank you.
Best Answer
A Jacobson ring is one where every prime ideal is the intersection of the maximal ideals which contain it. Examples include rings which are finitely generated over a field. For such rings one can define the Zariski topology just in terms of maximal ideals, and hence visualise closed subsets of affine space by the points they contain. Much of scheme theory can be developed in this setting, and there are various notes online doing this.