What is the difference between using maximal ideals to define Zariski topology versus using prime ideals?

Question

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.

Answer 1

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.

Answer 2

If You consider a homomorphism of rings $f:A\rightarrow B$ You want a corresponding mapping of $Specm B$ into $Specm A$ but for a maximal ideal $\mathfrak{b}\subset B$ the ideal $f^{-1}(\mathfrak{b})$ is not necessarily maximal. This is not the case if You consider prime ideals.

Answer 3

You do actually look at the max spec of rings. In classical algebraic geometry, we consider the ring $R = k[x_1, \dots, x_n]$ where $k$ is an algebraically closed field. Then we take $k^n$ and topologize it via the Zariski topology. By the Nullstellensatz, this is homeomorphic to MaxSpec(R) via the map $(a_1, \dots, a_n) \mapsto (x_1 - a_1, \dots, x_n - a_n)$.

There's also a more "intrinsic" characterization of Spec, which requires some technical knowledge to understand. In algebraic geometry, we look at locally ringed spaces, which turn out to be useful places for geometry to be done. There is a natural functor $LocRingSpaces \longrightarrow CommRings$ via $X \mapsto \mathcal O_X(X)$. Spec turns out to be adjoint to this functor, so by uniqueness of adjoints this can be treated as the definition of Spec.