The “Idea” of a Scheme

The mathematical concept of a “scheme” seems to pop up everywhere, but it’s hard to get a good grasp on what a scheme actually is. Any time you might ask someone what a scheme is in passing, there never seems to be enough time to explain it. On the other hand, if someone finds the time to internalize the full definition, it’s not immediately clear why a scheme is defined the way it is. The following interpretation helped me understand the idea of a scheme at a somewhat deeper level than a quick conversation — and hopefully it can help you too!

The idea of a scheme borrows from the definition of a manifold in differential topology. Recall that a manifold is a a topological space covered by charts diffeomorphic to open subsets of some Euclidean space $\mathbb{R}^n$. Now in a lot of ways, open subsets of $\mathbb{R}^n$ are “the place to do differential calculus” — for example, taking the derivative of a function on $\mathbb{R}^n$ at a point requires the function be defined on an open neighborhood about that point. In the same way, a scheme is a topological space* $X$  that locally looks like “the place to do algebra/find zeroes of functions”. However, “the place to do algebra/find zeroes of functions” is a long name, so we’ll just call this an affine scheme instead.

Okay, but what exactly is an affine scheme? Affine schemes are the full manifestation of a simple idea: If $A$ is a ring, we should view the elements of that ring as functions instead of points. If you’re thinking about rings like $\mathbb{Q}(i)$ or $\mathbb{Z}/6\mathbb{Z}$, this can be a little weird, but if you do the construction first imagining rings like $\mathbb{R}[x,y]$ it might be more clear. But the beautiful part of this is that this idea works for any ring — meaning that any ring can be viewed as functions on some space.

For the remainder of this article, fix a ring $A$. What space could this ring $A$ be functions on? To motivate the answer, let’s do an example from basic high school algebra. If you want to see where the polynomial $p(x) = x^3 + x^2 + x$ vanishes, your natural instinct would be to begin by factoring $p(x) = x(x^2 + x + 1)$. Then if $p(x) = 0$, you know that either $x = 0$ or $x^2 + x + 1  = 0$. Put in slightly more abstract terms, if the function $p(x)$ vanishes at some point, then either the function $x$ vanishes at that point or the function $(x^2 + x + 1)$ vanishes at that point. This is an important property we’d like our topological space $X$ to satisfy:

Property 1:  If $f, g \in A$ and the product $fg$ vanishes at a point $x \in X$, then either $f$ vanishes at $x$ or $g$ vanishes at $x$.

This definition looks suspiciously close to the definition of a prime ideal of a ring! Recall that a prime ideal $\mathfrak{p}$ of a ring is an ideal where if $f,g \in A$ and $fg \in \mathfrak{p}$, then either $f \in \mathfrak{p}$ or $g \in \mathfrak{p}$. So if we assume that the “points” of our special topological space are prime ideals of the ring $\mathfrak{p}$, then we have an obvious choice for what it might mean for a function $f \in A$ to vanish at the point $\mathfrak{p}$ — we simply say that $f$ is defined to vanish at $\mathfrak{p}$ if $f \in \mathfrak{p}$ (or equivalently, it is zero in the ring $A/\mathfrak{p}$).

Given a ring $A$, define the spectrum of $A$, written $\text{Spec}(A)$, to be (as a set) the set of prime ideals of $A$. Now, I’ve promised a topology on this space, and any good topology in a subject relating to “vanishing” should have the topology related to vanishing. And luckily for us, it does! We define a set of prime ideals (think “points”) to be closed if it is of the form $V(S) = \{\mathfrak{q} \in \text{Spec}(A) : S \subset \mathfrak{q}\}$, for any set of “functions” $S \subset A$ (meaning, “every element of $S$ vanishes at $\mathfrak{q}$). It’s not too hard to check this forms a topology on $\text{Spec}(A)$, and is a good exercise!

This is a good definition for an affine scheme (which, recall, was a place where we could talk about where functions vanish) because in some sense the definition is “maximal” — any set of elements in the ring $A$ that can be the set of functions vanishing at a point is a prime ideal (since it is subject to Property 1). Moreover, functions distinguish the points in the topological space — if two points $\mathfrak{p}, \mathfrak{q}$ in the space are distinct, then there is a function that vanishes on one of the points, but not on the other one. Seeing why this is true is a good check on one’s understanding of the concept. However, you don’t necessarily get to choose which point — keep reading!

Example: Consider the ring $A = \mathbb{C}[x,y]$. What are some points in $\text{Spec}(A)$? Certainly if one chooses some $a,b \in \mathbb{C},$ then $\mathfrak{p} = (x-a, y-b)$ is a maximal (hence prime) ideal, since the quotient $A/\mathfrak{p}$ is $\mathbb{C}$, which is a field. A good way to think about the point $(x-a, y-b)$ is literally to identify it with the point $(a,b) \in \mathbb{C}^2$. Now consider the function $p(x,y) = x^4 + x^3y-xy-y^2$, and take $a = 2, b = 8$. Then $p$ vanishes at $\mathfrak{p}$ since

$p(x) = (x^3 + 2x^2 + 4x + x^2y + 2xy + 4y)(x-2) + (-x-y)(y-8)$,

which shows that $p \in \mathfrak{p}$. Note that $p$ vanishes in the normal sense at $(2, 8)$ because $p(2, 8) = 2^4 + 2^6-2^4-2^6 = 0$!

Now, are there any other points in $\text{Spec}(A)$? The answer is, yes! The polynomial $q(x,y) = x^3-y$ is an irreducible polynomial by Eisenstein’s Criterion, so the ideal $\mathfrak{q} = (x^3-y)$ is another point in $\text{Spec}(A)!$ This has an obvious interpretation in $\mathbb{C}^2$ — it’s the curve $y = x^3$! And more amazingly, notice that we can factor

$p(x,y) = x^4 + x^3y-xy-y^2 = (x^3-y)(x + y)$,

so $p \in \mathfrak{q}$. The interpretation of this is very pretty — the function $p(x,y)$ vanishes on the entire curve associated to $q$. That is very neat.

This is about half of the definition of an affine scheme. Continuing with the “rings are functions on some natural space” interpretation, one develops a sheaf on the space $\text{Spec}(A)$ so that one can, for example, talk about the function $(x^2 + y^2)(x-3)$ on the places where $x-3$ doesn’t vanish. We won’t develop this here, but this article has ideally given you a good picture of what the topological space of an affine scheme looks like, and has given you a new interpretation of rings!

*Note: It’s technically a little more than that — it’s actually a ringed space, which we won’t really get into here. Basically, every possible function you can construct on the space is built into the definition with appropriate “restriction” maps.