The Algebra-Geometric Motivation Behind Considering The Prime Spectrum of Arbitrary Rings

As asked in the comments here is the explanation of geometric points and the explanation of how do you work with points in more general algebraic geometry.

Suppose $A$ is a $k$-algebra of finite type. Then $$A=k[X_1,\dots,X_n]/(f_1(X_1,\dots,X_n),\dots,f_k(X_1,\dots,X_k))\tag{$\star$}$$ for some polynomials $f_1,\dots,f_k$.

If $k$ is algebraically closed then the points of the algebraic set $V(f_1,\dots,f_k)\subseteq \mathbb{A}^n$ can be recovered from $A$ by looking at its maximal ideals. This is part of what Hilbert‘s Nullstellensatz says.

Nevertheless, there is another way of obtaining the points of $V(f_1,\dots,f_k)\subseteq \mathbb{A}^n$ from the ring $A$. This other way happens to be much easier (it is kinda tautological) and much more general than the previous one (although when $k$ is alg. closed they are equivalent of course) and goes as follows: The points of $V(f_1,\dots,f_k)$ are exactly the morphisms of $k$-algebras from $A$ to $k$. This happens because having a morphism of $k$-algebras $$\overline{\phi}:A\rightarrow k$$ is the same as having a morphism of $k$-algebras $$\phi:k[X_1,\dots,X_n]\rightarrow k$$ that vanishes over $f_1,\dots,f_k$. And by the universal property of the polynomial algebra, this morphism is given by fixing the values for $X_1,\dots,X_n$. That is, is given by choosing elements $b_1,\dots,b_n\in k$ such that $f_i(b_1,\dots,b_n)=0 \ \forall i$. Hence we conclude that $$V(f_1,\dots,f_k)=\operatorname{Hom}_{\text{$k$-alg}}(A,k).$$

But now notice the following:

• We didn't use the fact that $k$ is algebraically closed, actually, we didn't use the fact that it is a field. So we can take it to be a ring.
• We can use the same trick to find the solutions of the system of equations $\{f_1=0,\dots,f_k=0\}$ over any $k$-algebra. The set of all such solutions is the set of $k$-algebra morphisms $\operatorname{Hom}_{k\text{-alg}}(A,B)$.

Now this is the way which generalize to algebraic geometry over any base ring $k$. The idea is that now instead of working directly with the algebraic set $V(f_1,\dots,f_k)\subseteq \mathbb{A}^n$ you will work with the "$k$-scheme" $\operatorname{Spec}(A)$, and if you want to recover $V(f_1,\dots,f_k)$ you just have to look at the set of all morphisms $$\operatorname{Hom}_{k\text{-schemes}}(\operatorname{Spec}(k),\operatorname{Spec}(A))$$ because this set will happens to be (by definition) equal to $$\operatorname{Hom}_{\text{$k$-alg}}(A,k)$$ and we know that these are the solutions of the system of equations. Hence, from the scheme you get the solutions to the system of equations as morphisms. They are NOT the points of the underlying set of the schemes (this is one of the difficulties that people encounter while learning about schemes). These other points are prime ideal ideals that are not longer directly related with the solutions of the system of equations anymore.

If $X$ is your scheme, in order to make a clear disctinction. The points in the underlying set of $X$ are called schematic points of $X$, and this are part of the engines for making $X$. The set of all morphisms $\operatorname{Hom}_{k\text{-schemes}}(\operatorname{Spec}(k),X)$ is called the set of $k$-geometric points of $X$ and is denoted by $X(k)$. More generally, for any $k$-algebra $B$ the set of $B$-geometric points of $X$ is $\operatorname{Hom}_{k\text{-schemes}}(\operatorname{Spec}(B),X)$ and is denoted by $X(B)$ (although most of the time it is customary to restrict geometric points to the case in which $B$ is a field).

Example: Take $X=\operatorname{Spec}(\mathbb{Z}[X,Y,Z]/(X^2+Y^2-1))$, then

• The set of schematic points of $X$ are the prime ideals of $\mathbb{Z}[X,Y,Z]/(X^2+Y^2-1)$ and this sounds difficult to compute...

• The set $X(\mathbb{Z})$ of $\mathbb{Z}$-geometric points is $(X,Y)=\{(\pm 1,0), (0,\pm 1)\}$.

• As $\mathbb{Q}, \mathbb{R}$ are $\mathbb{Z}$-algebras we can also compute the set of geometric points over this fields and we get $X(\mathbb{Q})=\{(\frac{2mn}{m^2+n^2},\frac{m^2-n^2}{m^2+n^2}); m,n\in \mathbb{Z} \text{ coprime}\}$ and $X(\mathbb{R})$ is the circle.

• As $\mathbb{Z}[X]$ is also a $\mathbb{Z}$-algebra we can again compute the geometric points $X(\mathbb{Z}[X])$ and they correspond to morphisms between the integral affine line ($\operatorname{Spec}(\mathbb{Z}[X])$ is the integral affine line) and the integral circle. This shows that, with the generality we are working on, morphisms between algebraic varieties fall in the same family as points and a lot of times you will be able to apply theorems about points to morphisms.

One aspect even of classical algebraic geometry over an algebraically closed field $k$ is the following: the aim of algebraic geometry is not "just" solving systems of polynomial equations but to understand the structure of the resulting set of solutions. This is directly comparable to linear algebra: one does not just solve systems of linear equations but considers the set of solutions as a vector- resp. affine space.

In algebraic geometry considering the structure of an algebraic variety $X$ means that one also wants to understand the nature and the relation of the various subvarieties of $X$ to each other. This information is encoded in the prime spectrum of the coordinate ring $k[X]$ of $X$.

In the immediate predecessors to modern algebraic geometry as formulated for example by Van der Waerden this approach was indeed expressed in terms of solving polynomial equations - but with coefficients in a so-called universal field extension $\Omega$ of $k$: adjoin countably many algebraically independent elements $x_i$, $i\in\mathbb{N}$, to $k$ and take the algebraic closure of the resulting rational function field. This approach was given up in favor of scheme theory.

I am the wrong person to describe the details of the theory/process, but I wan't to bring up one aspect.

Spectra and schemes allow us to move from characteristic zero to characteristic $p$.

When a geometric object can be described with polynomial equations with integer coefficients, then the machinery allows us to bring the topology and differential geometry into the picture by extending the ring of scalars from $\Bbb{Z}$ to $\Bbb{C}$ using the inclusion. And it allows us to go from $\Bbb{Z}$ to $\Bbb{F}_p$ by reducing modulo $p$. We can then entertain hopes of bringing geometrical intuition into positive characteristic problems, and that has turned out to be fruithul.

The crowning achievement here is the handling of Weil conjectures that, among other things, connect cohomology groups of basic algebraic topology to the number of solutions of systems of polynomial equations.

This program lead to the development of powerful theories. Alas, I'm too ignorant to describe those ideas well.

I'm more familiar with two problem areas, where similar thinking has been used successfully. Listing them here:

• The representation theory of simple algebraic groups has used this a lot. The characteristic zero theory (representations of simply Lie groups) linearizes well, and can be handled with the machinery of Lie algebras alone. Characteristic $p$ is more difficult. But, the groups (or group schemes) are defined over $\Bbb{Z}$, so the above grand approach can be applied- Rewrite and reprove characteristic zero results (possibly originally proved using tools from analysis) in a language that can be ported to $\overline{\Bbb{F}_p}$ via $\Bbb{Z}$. Study what survives, what changes, and how. There was a lot of work going on there in the 80s (give or take).
• In coding theory some aspects of the geometry of curves lead to the study of so called algebraic geometry codes or Goppa codes. Cutting a longer story short: Riemann-Roch tells us that certain kind of large codes exist, and Weil conjectures (here called the Riemann hypothesis of function fields) relate parameters of those codes leaving researchers an interesting game to play.

Of course, the connections to number theory, arithmetic geometry in particular, is an area of very active research. Unfortunately nearly all of it is beyond me. We have users working in the area, and hopefully they can answer this better.