Some notes on varieties and ideals

Simply put, the goal of algebraic geometry is the study the geometry of the zeros of polynomial equations with algebra. As such, we have two perspectives of viewing “zeros of a polynomial equation” and mathematical objects to encapsulate them. Classically, encapsulating the geometric perspective are affine varieties:

DEFINITION. Let $K$ be a field and $f_1, \dots, f_s$ be polynomials in $K[x_1, \dots, x_n]$. Then the affine variety defined by $f_1, \dots, f_s$ is defined as $\mathcal V(f_1, \dots, f_s) = \{(a_1, \dots, a_n) \in K^n : f_i(a_1, \dots, a_n) = 0 \text{ for all } 1 \leq i \leq s\}.$

In other words, affine varieties are the set of points on which all the polynomials $f_1, \dots, f_s$ simultaneously vanish once evaluated at those points. This is a geometric perspective in the sense that we’re focusing on a subset of points in some affine space $K^n$, which is a generalization of $n$-dimensional Euclidean space we (and the ancient Greeks) first think of as “geometry”.

But what if we flip this question around? Instead of the points at which some polynomials vanish, what if we focus on what we can say about the polynomials themselves, that is, take a more “algebraic” perspective? Well, it turns out we can say a lot, in particular because these polynomials exist in a ring studied in abstract algebra, namely the polynomial ring $K[x_1, \dots, x_n]$. And it would nice if we could translate what we can say about polynomial rings into something geometric about affine varieties–if for nothing more than to find a more visual representation of the algebraic objects we’re working with.

To bridge the gap, we might turn to certain subsets of (in this case, polynomial) rings, called ideals, that turn out to be ideal for the job. I’ll start with a definition and then note why they can be useful.

DEFINITION. A subset $I \subseteq K[x_1, \dots, x_n]$ is an ideal if it satisfies

1. $0 \in I$.
2. If $f, g$, then $f + g \in I$.
3. If $f \in I$ and $h \in K[x_1, \dots, x_n]$, then $hf \in I$.

So ideals are sets of polynomials that are closed under addition and absorb multiplication. You might notice this means ideals are also closed under “polynomial combinations” like subspaces in linear algebra, so we can naturally define an ideal generated by a few basis polynomials analogous to linear span:

DEFINITION. Let $f_1, \dots, f_s$ be polynomials in $K[x_1, \dots, x_n]$. Then the ideal generated by the basis $f_1, \dots, f_s$, denoted $\langle f_1, \dots, f_s \rangle$, is defined as $\langle f_1, \dots, f_s \rangle = \left\{ \sum_{i=1}^s h_if_i : h_i \in K[x_1, \dots, x_n] \right\}.$

But how does this help us cross into the land of geometry and affine varieties? Well, there’s actually a nice way to think of $\langle f_1, \dots, f_s \rangle$ that has an varietal “flavor”. Construct the system of equations: $\begin{array}{c} f_1 = 0, \\ f_2 = 0, \\ \vdots \\ f_s = 0. \end{array}$ Now if this system has a solution, we’ve got ourselves a point on the variety $\mathcal V(f_1, \dots, f_s)$. Further, if we whip up any polynomial combination for $h_i \in K[x_1, \dots, x_n]$, then evaluated at that solution, $h_1f_1 + h_2f_2 + \cdots + h_sf_s = 0.$ Yet this is exactly an element of the ideal $\langle f_1, \dots, f_s \rangle$! So the ideal $\langle f_1, \dots, f_s \rangle$ can be thought of as all the other polynomials that also vanish¹ if $f_1, \dots, f_s$ all vanish there. In fact, an important result, Hilbert’s Basis Theorem states that any ideal $I$ of $K[x_1, \dots, x_n]$ can be generated by a finite basis, so if any ideal vanishes on its basis, it consists of all the other polynomials that vanish as well.

Thus we seem to have two questions here. First, “given a set of polynomials, at which points do those polynomials all vanish?” and the reverse “given a set of points, which polynomials vanish at those points?” The former is answerable by varieties where the given “set of polynomials” is a given ideal:

DEFINITION. Let $I \subseteq K[x_1, \dots, x_n]$ be an ideal. Then we define the variety of ideal $I$ as $\mathcal V(I) = \{(a_1, \dots, a_n) \in K^n : f(a_1, \dots, a_n) = 0 \text{ for all } f \in I\}.$

and the latter is answerable by ideals where the given “set of points” is a given variety:

DEFINITION. Let $V \subseteq K^n$ be an affine variety. Then we define the ideal of variety $V$ as $\mathcal I(V) = \{f \in K[x_1, \dots, x_n] : f(a_1, \dots, a_n) = 0 \text{ for all } (a_1, \dots, a_n) \in V\}.$

So $\mathcal V$ maps ideals to varieties and $\mathcal I$ varieties to ideals. This could be our link between geometry and algebra, and we’d love if $\mathcal V$ and $\mathcal I$ were exactly inverses of each other, allowing us to easily interface algebraic and geometric results! But are they?

Sadly, no.

But this question gets at the heart of incredibly important results such as Hilbert’s Nullstellensatz and allows us to pinpoint exactly what types of ideals have this nice invertible correspondence with varieties–radical ideals. In the process, we’ll see that, unlike linear algebra where we didn’t much about the field we worked in to solve linear systems, algebraic geometry cares a huge deal about the field we work in to solve polynomial systems.