Layman's Question on Schemes

Question

I am reading Jordan Ellenberg's article on Arithmetic Geometry in the Princeton Companion to Mathematics. I have forgotten most of the algebra I learned since passing my qualifying exams more than 30 years ago. The article is beautifully written and I followed almost everything he said and how schemes is a way to associate a geometric object with any ring in Section 3.3 on page 377. In section 3.4, he gave the example of $\text{Spec } \mathbb Z$ and that each point $n$ of $\mathbb Z$ can be consider as a map from $\text{Spec } \mathbb Z$ to $\mathbb C$ using $n\mod p$. Great. It works. My question is how is this done in general for an arbitrary ring $R$? (The only way I can think of is to use the quotient for each prime ideal $(p)$ and send $(p)$ to $n+(p)$. However, this would mean that the range of map changes with the argument $(p)$. The problem I have with that is that for a fixed $n\in R$, the range of $n$ considered as a map changes with the argument $(p)$.) Another question: what is a good reference to read a little more on the topic without trying to become an expert in the area.

Answer 1

You're right that the range of the map you want to write down changes with $(p)$. In fact this happens even in the case of $\mathbb Z$: Ellenberg doesn't have space to write down many details, but the map he describes is $\mathbb Z\to \mathbb Z/(p), n\mapsto n\pmod p$. Why this map, rather than the one to $\mathbb C$ you suggested? Simply because the latter map is not a ring homomorphism, so doesn't reveal the structure these maps are supposed to reveal.

In general, for a commutative ring $A$ there is a "map" associating to each $x\in A$ its "value" at the prime ideal $\mathfrak p.$ (Here I write $\mathfrak p$ instead of $(p)$ because $\mathfrak p$ might not be generated by one element for a general $A$.) This simply takes $x$ to $x\pmod{ \mathfrak{p}}$, that is, to the value of $x$ in the quotient ring $A/\mathfrak p$. We can see this agrees with the example of $\mathbb{Z}$. For one more point on this: in classical algebraic geometry over, most often, $\mathbb C$, our $A$ becomes a ring of polynomials $\mathbb C[X_1... X_n]$ and these "evaluation" homomorphisms correspond to the actual evaluation of a polynomial at a point in $\mathbb C^n$. In general it would be nice if our evaluations at least landed in a field, to salvage some of the computational facility of the classical situation. For this one often takes the evaluation's range as the field of fractions $\kappa_\mathfrak p$ of $A/\mathfrak p$ instead of $A/\mathfrak p$ itself.

As to learning more: you might look at this exposition of some of the justification for beginning to work with schemes. The first book I'd recommend is The Geometry of Schemes by Eisenbud and Harris. If you don't remember much about sheaves, or about commutative algebra, it might be tough going to jump into general schemes, in which case you might page through Eisenbud's Commutative Algebra with a View Towards Algebraic Geometry. This is basically a book on affine schemes, that is, the spectra of commutative rings, and has many very pleasant motivating examples.