Categorical definition of Spectrum of a ring as a colimit?

Question

I've been watching AGITTOC: Algebraic geometry in the time of Covid; Pseudolecture 3, where a user describes an alternative definition of the spectrum of a ring:

(Name hidden for privacy): I like the definition of $\operatorname{Spec}(A)$ that doesn't include the word prime ideal, by a colimit of $\operatorname{Hom}(A, k)$ where $k$ run over all fields and the maps are morphisms that make the diagrams commute.

I've been trying to find a reference to this, to no avail. The closest thing could find is this reference in the Stacks project which talks about how if the ring $A$ is built up as a colimit of $A_i$, then the spectrum $\operatorname{Spec}(A)$ is built up as the limit of $\operatorname{Spec}(A_i)$. This does not seem to be what I am looking for.

Can someone please provide a reference and/or quickly exposit this definition of $\operatorname{Spec}(A)$ which I have not seen before?

Answer 1

The main idea there is that if $f:A\to k$ is a morphism of rings, then $\ker f$ is a prime ideal of $A$.

I would like to point out that this is precisely how Peter Scholze defines the spectrum of a ring in his Bonn course. There are some typed notes here: https://www.math.uni-bonn.de/people/ja/alggeoI/notes.pdf.

Answer 2

The key word to look for here is "functor of points", and the key observation is that the category of affine schemes is equivalent to the opposite category of the category of commutative rings. This is often stated as a theorem involving $Spec(R)$ as a locally ringed space, but once one knows this, you can use whatever model of $Ring^{op}$ you desire, since they give the exact same category. I feel like this should be emphasised more, there are many ways of realising this category as something concrete, but they are all the same thing.

The (usual) locally ringed spaces approach starts with this "geometric" seeming definition using prime ideals and structure sheaves, and then it becomes a nontrivial theorem that this yields $Ring^{op}$.

The functor of points approach defines an affine scheme as a representable functor $X:Ring\rightarrow Set$, and we see the category of these is equivalent to $Ring^{op}$ by the Yoneda embedding. So this way we avoid the difficulty of showing that $Aff\cong Ring^{op}$, at the cost of somewhat opaque geometry.

What I believe the OP is referring to is that one may show that any representable functor $Hom(R,\_)$ is determined by less data than its inputs on all rings, and we can get by using only fields $k$. I haven't checked this, but it shouldn't be too hard to prove if true.

There is an extended discussion on precisely these things here: https://sbseminar.wordpress.com/2009/08/06/algebraic-geometry-without-prime-ideals/

You can find a nice exposition of the functor of points approach in Eisenbud-Harris' Geometry of Schemes.