TITLE: MaxSpec, Spec, ... "RadSpec"? Or, why not look at all radical ideals?
QUESTION [27 upvotes]: I was reading this question on why algebraic geometry looks at prime ideals instead of only maximal ideals, and I understand Anton's answer, but I'm a little confused as to how this fits with Hilbert's Nullstellensatz - affine algebraic sets are in bijection with radical ideals, not prime ideals, and it seems like we'd want the extra information we'd get by looking at "RadSpec(R)" (my own imagined notation). Also, the preimage of a radical ideal is radical, so there isn't the same objection as to maximal ideals - "RadSpec" would also be a contravariant functor.
So, why not radical ideals instead of only prime ideals, and what kinds of things could we say about RadSpec(R) even if they aren't very interesting?

REPLY [3 votes]: The category of reduced $k$-algebras over a field $k$ is an example of a so-called Zariski category; see the book "Categories of commutative algebras" by Yves Diers. In every Zariski category, basic notions of commutative algebra can be developped and over every Zariski category $A$ we can consider the category of $A$-schemes. The theory shows that, in fact, most of the basic algebraic geometry takes over to this general setting. And if $A$ is a so-called rational reduced Zariski category, it turns out that the basic notions of varieties take over (such as birational equivalence is detected by the function field object, and Hilbert's Nullstellensatz). If $A$ denotes the category of reduced $k$-algebras, we get the usual category of reduced $k$-schemes. So in some sense, this is an algebraic geometry in its own right which you can study. The functor of points approach also works with functors $A \to \text{Sets}$. Note, however, that here the spectrum of a reduced $k$-algebra also consists of the usual prime ideals, not all radical ideals. Nevertheless, you may read Section 2.4 in Dier's book. There radical congruences in Zariski categories are studied in general.