Algebra and Geometry are opposites

The title is a bit of a clickbait (though it’s not false per se): it’s probably more correct to say that algebra and geometry are dual. This is a pretty striking phenomenon, and I am not well-enough equipped to really discuss it, but that’s not going to stop me from trying. Perhaps, at the very least, it’s not terribly difficult to see that there’s some connection between algebra and geometry. Given an algebraic equation (or even a system thereof), we can produce something geometric by plotting the set of solutions. For example: consider the equation $x^2+y^2+z^2=1$ . If you plot out all the (real) values $(x,y,z)$ that solve this equation, you get the unit sphere!

You can also get a whiff of the duality: if we add more equations, we get a smaller geometric set. If we took our unit sphere (corresponding to $x^2+y^2+z^2=1$ ) and assert the additional constraint that $z=0$ , then we have removed the third dimension of our space and we are left with a 2D space of solutions $(x,y)$ to $x^2+y^2=1$ —this is the unit circle lying at the equator of our sphere!

Gelfand duality

If our geometric object is not given as the solution set of a system of algebraic equations, there is still a way (in fact, a more generally applicable way) to produce something algebraic. Given some geometric space $X$ (more precisely, it needs to be compact and Hausdorff for the duality to happen), we can consider the set $C(X)$ of continuous functions $X\to\mathbb{C}$ into the field of complex numbers (with its usual topology). The complex numbers carries a lot of algebraic structure: in particular, we can add and multiply them together. This structure is inherited by $C(X)$ pointwise: given functions $f,g:X\to\mathbb{C}$ , we can add them together with $(f+g)(x) := f(x)+g(x)$ , and likewise multiply them together with $(f\cdot g)(x) := f(x)\cdot g(x)$ . In fact, $C(X)$ inherits many other properties from $\mathbb{C}$ :

We can rescale a function $f:X\to\mathbb{C}$ by a complex number $\lambda\in\mathbb{C}$ by setting $(\lambda f)(x) := \lambda f(x)$ . This, with the addition and multiplication of functions, makes $C(X)$ into an (associative, unital) algebra over $\mathbb{C}$ .
We can give a function $f:X\to\mathbb{C}$ a norm (i.e., absolute value) $||f|| := \sup_{x\in X}|f(x)|$ . This endows $C(X)$ with a topology (and in particular a means to talk about limits), and we find that $C(X)$ inherits completeness with respect to this norm from the usual completeness of $\mathbb{C}$ . In fact, the norm makes $C(X)$ into a Banach algebra.
Finally, complex conjugation in $\mathbb{C}$ gives us an involution on $C(X)$ which transforms a function $f:X\to\mathbb{C}$ into the function $f^*:X\to\mathbb{C}$ given by $f^*(x) := \overline{f(x)}$ . This involution ultimately makes $C(X)$ into what is called a C*-algebra.

We can see the duality again: if $F:X\to Y$ is a continuous function (of compact Hausdorff spaces), then we get a map $F_*:C(Y)\to C(X)$ (in the other direction!) that sends a continuous function $f:Y\to\mathbb{C}$ to the composite $F_*(f) := f\circ F : X\xrightarrow FY\xrightarrow f\mathbb{C}$ . Moreover, this map is a *-homomorphism (meaning it preserves the C*-algebra structure).

In categorical terms, $C$ defines a functor $\mathbf{CHaus}^{\mathrm{op}}\to\mathbf{C^*Alg}$ from the opposite of the category of compact Hausdorff spaces (and continuous functions) to the category of C*-algebras (and *-homomorphisms). What’s striking (and establishes the “duality” here to the full extent) is that $C$ is an equivalence of categories!

Theorem 1 (Gelfand duality). The category of compact Hausdorff spaces is formally dual to the category of C*-algebras.

To go in the other direction, you would have to associate to every C*-algebra $A$ a space $\hat A$ . The trick here is to take advantage of the fact that $\mathbb{C}$ also has canonical geometric properties, and define $\hat A$ to be the set of nonzero $\mathbb{C}$ -algebra homomorphisms $\chi:A\to\mathbb{C}$ (called characters). Then, endow $\hat A$ with the weak-* topology (which makes functions converge if they converge pointwise; i.e., $\lim_{n\to\infty}\chi_n=\chi$ iff $\lim_{n\to\infty}\chi_n(a) = \chi(a)$ for all $a\in A$ ). This will be a compact Hausdorff space, and $A\cong C(\hat A)$ by sending $a\in A$ to the function $\hat A\to\mathbb{C}, \chi\mapsto \chi(a)$ . This is called the Gelfand dual of $A$ .

Side note: There is an equivalent way of constructing the Gelfand dual: we can look at its primitive spectrum. Given a C*-algebra $A$ , a primitive ideal of $A$ is the kernel of an irreducible *-representation. The set of primitive ideals inherits a canonical topology by taking the limit points of a set $S$ of primitive ideals to be those primitive ideals $\mathfrak p$ such that $\mathfrak p\supseteq\bigcap_{\mathfrak q\in S}\mathfrak q$ . Under this topology, we have a homeomorphism $\hat A\cong\mathop{\mathrm{Prim}}(A)$ by sending a character in $\hat A$ to its kernel.

We have actually seen a similar duality between spaces and algebras before (well… assuming you read my blog haha). In some sense, you can think of a set of points as the most unstructured kind of geometric object, and so it should correspond to some sort of dual algebraic notion. Indeed, we worked out that said dual notion is the category of complete atomic Boolean algebras. Moreover, the construction of the algebra from a set $X$ was given by looking at the algebra of functions $X\to\{0,1\}$ , where $\{0, 1\}$ carries canonical Boolean algebra structure!

A variety of this duality

Let’s return to the example with circles and spheres (more specifically, solution sets of systems of polynomial equations). We can actually recover algebraic structure from these geometric objects in a similar manner to the above (I am referring to the construction $X\mapsto C(X)$ ), but instead of continuous functions into $\mathbb{C}$ , we should think about “polynomial” functions into $\mathbb{C}$ . To illustrate what I mean, let $V := \{(x,y,z) \mid x^2+y^2+z^2=1\}$ be the unit sphere, then a “polynomial” function $f:V\to\mathbb{C}$ should essentially be the restriction of a three-variable polynomial $f(x,y,z)$ (with complex coefficients) to $V$ . For example, we could take $f(x,y,z) = xyz$ , or we could take $g(x,y,z) = 1+x+yz^2$ , or we could also take $h(x,y,z) = xyz(x^2+y^2+z^2)$ .

However, you might notice that $f|_V=h|_V$ . Indeed, $x^2+y^2+z^2=1$ in $V$ by definition, so the expression for $h$ reduces to be the same as the expression for $f$ when we restrict to $V$ . Therefore, they should be considered the same. More generally, two polynomials $f,g\in\mathbb{C}[x,y,z]$ should be considered the same if their difference $f-g$ is a multiple of $x^2+y^2+z^2-1$ (since this polynomial is “equal to zero” on $V$ ). This is just an explicit way of saying that the ring of “polynomial” functions $V\to\mathbb{C}$ (which is actually called the ring of regular functions on $V$ ) is given by the quotient ring $\mathbb{C}[V] = \frac{\mathbb{C}[x,y,z]}{x^2+y^2+z^2-1}$ . This ring is also called the coordinate ring of $V$ .

More generally, these kinds of spaces are called affine varieties: more concretely, an affine variety is the zero locus of a family of polynomials $f_1,\dots,f_m \in \mathbb{C}[x_1,\dots,x_n]$ —meaning that an affine variety is given by the subset $V(f_1,\dots,f_m) \subseteq \mathbb{C}^n$ of those points $(x_1,\dots,x_n)$ where $f_i(x_1,\dots,x_n) = 0$ for every $i=1,\dots,m$ —where the family of polynomials is “reduced”. To see what I mean by “reduced”, let me give an example of something that is not. Recall that we could carve out the equator of our sphere by adding the constraint $z=0$ , so that our circle is given by $V(x^2+y^2+z^2-1, z)$ . We could have also obtained the equator by instead adding the constraint $z^2=0$ (giving $V(x^2+y^2+z^2-1, z^2)$ ), but this doesn’t seem as “reduced” since $z^2=0\implies z=0$ .

To be more precise, by “reduced” I mean that the family of polynomials generates a radical ideal. The reason for restricting to radical ideals is twofold. First, there is no loss of generality: if the generated ideal $I$ is not a radical ideal, then just take its radical $\sqrt{I}$ , then $V(f_1,\dots,f_m) = V(I) = V(\sqrt{I})$ . Secondly, and more importantly, this makes constructing the coordinate ring of $V=V(I)$ easier. Indeed, with $I$ being a radical ideal, we can actually say that two polynomial functions $f,g:V(I)\to\mathbb{C}$ are equal as regular functions of $V(I)$ if and only if their difference $f-g$ lies in the ideal $I$ . (This is the content of Hilbert’s Nullstellensatz.) Therefore, if $V=V(\sqrt I)$ is an affine variety, then its coordinate ring is given by $\mathbb{C}[V] = \frac{\mathbb{C}[x_1,\dots,x_n]}{\sqrt I}$ .

To see how this gives us a Gelfand-esque duality, we need to describe the appropriate categories. On one hand, rings of the form $\frac{\mathbb{C}[x_1,\dots,x_n]}{\sqrt I}$ are precisely the finitely-generated reduced $\mathbb{C}$ -algebras, so the natural choice of category is the category $\mathbb{C}\mathbf{Alg}_{\mathrm{red}}$ of finitely-generated reduced $\mathbb{C}$ -algebras and $\mathbb{C}$ -algebra homomorphisms.

On the other hand, a morphism of affine varieties $f:V\to W$ where $V\subseteq\mathbb{C}^n$ and $W\subseteq\mathbb{C}^m$ should be a polynomial function $f:\mathbb{C}[x_1,\dots,x_n]\to\mathbb{C}[y_1,\dots,y_m]$ which restricts to a function $V\to W$ (in that $f(x_1,\dots,x_n)\in W$ if $(x_1,\dots,x_n)\in V$ ). However, we have the same caveat as for regular functions: two such morphisms $f,g:V\to W$ should be considered the same if they agree on all values of $V$ (even if they come from different polynomials). This describes the category $\mathbf{Aff}_{\mathbb{C}}$ of affine varieties and their morphisms. In particular, if $\mathbb{A}^1_{\mathbb{C}} = V(\varnothing) = \mathbb{C}$ is the complex plane viewed as an affine variety, then morphisms $f:V\to\mathbb{A}^1_{\mathbb{C}}$ are precisely the regular functions on $V$ .

Notice how a morphism of affine varieties $f:V\to W$ induces a homomorphism of coordinate rings $f_*:\mathbb{C}[W]\to\mathbb{C}[V]$ (in the other direction!) by sending a regular function $g:W\to\mathbb{A}^1_{\mathbb{C}}$ to $f_*g := g\circ f : V\xrightarrow fW\xrightarrow g\mathbb{A}^1_{\mathbb{C}}$ . Therefore, the construction of coordinate rings defines a functor $\mathbb{C}[-]:\mathbf{Aff}_{\mathbb{C}}^{\mathrm{op}}\to\mathbb{C}\mathbf{Alg}_{\mathrm{red}}$ .

Theorem 2. The functor $\mathbb{C}[-]:\mathbf{Aff}_{\mathbb{C}}^{\mathrm{op}}\to\mathbb{C}\mathbf{Alg}_{\mathrm{red}}$ is an equivalence of categories.

We have actually already talked about how to go in the other direction. Any finitely-generated reduced $\mathbb{C}$ -algebra can be written in the form $\frac{\mathbb{C}[x_1,\dots,x_n]}{\sqrt I}$ essentially by definition. This gives us, in particular, an ideal $I$ of polynomials. Thus, take our affine variety to be the space $V := V(I) \subseteq \mathbb{C}^n$ ; that is, the zero locus of the polynomials generating $I$ . The story is exactly the same if you replace $\mathbb{C}$ by any algebraically closed field.

Side note: Similar to the Gelfand dual of a C*-algebra, we can construct an affine variety from a finitely-generated reduced $\mathbb{C}$ -algebra $A$ by looking at its maximal spectrum. Note that points of an affine variety $V$ correspond to maximal ideals of its coordinate ring $\mathbb{C}[V]$ . Indeed, $x\in V$ gives us the maximal ideal $\mathfrak{m}_x$ generated by the regular functions $f:V\to\mathbb{A}^1_{\mathbb{C}}$ where $f(x)=0$ . Conversely, given a maximal ideal $\mathfrak{m}$ of $\mathbb{C}[V]$ , it turns out that there is a unique point $x\in V$ such that $f(x)=0$ for every $f\in\mathfrak{m}$ . This explains how to go backwards: given a finitely-generated reduced $\mathbb{C}$ -algebra $A$ , we can define an affine variety $\mathop{\mathrm{Spec}}_{\mathrm m}(A)$ whose points are precisely the maximal ideals of $A$ . This is a “basis-free” (in the sense of linear algebra) way of constructing the affine variety associated to $A$ . (Technically, you need more data than just this set of points for this to be fully formal—see the next section.)

A fine scheme for constructing duals

Jumping between algebra and geometry can be extremely powerful and helpful when doing mathematics: in the world of algebra, everything is fully formal and axiomatised, so it’s easier to write rigorous proofs; in the world of geometry, everything is spatial, so it’s easier to intuit and reason with these objects. Intuition is a great asset (except when it isn’t), and since abstract algebra is generally hard, this really encourages trying to give all algebras a geometric dual. If we relax our finitely-generated reduced $\mathbb{C}$ -algebra so that it is no longer finitely generated, no longer reduced, and no longer an algebra over $\mathbb{C}$ , then can it still be thought of as the coordinate ring of some affine variety? Well…. obviously not, but this is where we use the old French trick of turning theorems into definitions:

Definition 3: Define the category of affine schemes to be $\mathbf{Aff} := \mathbf{CRing}^{\mathrm{op}}$ , the opposite of the category of commutative rings and ring homomorphisms.

Now, given a ring $R$ , write $\mathop{\mathrm{Spec}}(R)$ for its corresponding affine scheme; conversely, given an affine scheme $X$ , write $\Gamma(X)$ for its corresponding ring (and call this its coordinate ring). By design, this formally makes every ring a “ring of regular functions” for some “space”, but this isn’t particularly helpful unless we can give the geometric side some actual meat. To do so, we need to revisit our genuine affine varieties and look more closely at its geometry.

Given a radical ideal $I$ of $\mathbb{C}[x_1,\dots,x_n]$ , I threw under the rug the geometry of $V=V(I)$ . Since it was a subset of $\mathbb{C}^n$ (which we can think of as $2n$ -dimensional real space), it’s already believable that it should be geometric. However, the topology on $V$ should not be the one inherited from the canonical Euclidean topology on $\mathbb{C}^n$ : the usual topology doesn’t tell us too much about $V$ as an algebraic set, and it doesn’t generalise well to when we replace $\mathbb{C}$ with another (algebraically closed) field.

We rectify this by using the Zariski topology, which is a completely algebraically-flavoured topology: define the closed sets of $V$ to be precisely zero loci for families of polynomials (regular functions) over $V$ . Equivalently, the Zariski topology is generated by basic open sets of the form $D(f) := \{x\in V \mid f(x)\neq0\}$ for $f:V\to\mathbb{A}^1_{\mathbb{C}}$ a regular function. This is the smallest topology on $V$ that ensures that (a) polynomials (regular functions) are continuous, and (b) singleton points $\{x\}\subseteq V$ are closed, so in some sense the Zariski topology is the best shot we have at encoding algebraic structure with topological means.

Side note: While this certainly makes $V$ a topological space, it’s probably to some extent dishonest if I claim this allows us to intuit about $V$ using our inherent human capacity to reason with spatial things. I mean, this topology is rather pathological, given that it’s not even Hausdorff (unless the variety was already a finite set).

The Zariski topology also allows us to be more nuanced with how we probe the affine variety with polynomials. The open set $D(f)$ is the subset of $V$ where $f$ does not vanish, so in particular we are allowed to “invert” $f$ on this set; that is, we can make sense of $\frac1f$ here. Therefore, the ring of regular functions on $D(f)\subseteq V$ is given by $\mathbb{C}[V][f^{-1}]$ (that is, the ring obtained by adjoining a formal inverse of $f$ ). Since we are now talking about regular functions defined “locally” on $V$ , this upgrades our coordinate ring on $V$ to a sheaf of regular functions $\mathcal{O}_V$ , where the local sections on $D(f)$ are given by $\mathcal{O}_V(D(f)) := \mathbb{C}[V][f^{-1}]$ . (Good thing I know what a sheaf is…)

Now we can redefine a morphism of affine varieties to be a continuous function $f:V\to W$ (under the Zariski topology) along with a morphism of sheaves $f^\sharp : \mathcal{O}_W\to f_*\mathcal{O}_V$ (in the other direction!). Indeed, the idea is that $f^\sharp$ sends a regular function $g:U\to\mathbb{A}^1_{\mathbb{C}}$ (for some open $U\subseteq W$ ) to the regular function $f^\sharp g := g\circ f : f^{-1}(U) \xrightarrow fU \xrightarrow g\mathbb{A}^1_{\mathbb{C}}$ . This is all we need to fully characterise affine varieties!

In particular, this allows us to make sense of the side remark in the previous section. Given a finitely-generated reduced $\mathbb{C}$ -algebra $A$ , we can translate the Zariski topology and the sheaf of regular functions to the abstractly-defined affine variety $\mathop{\mathrm{Spec}}_{\mathrm m}(A)$ . The idea of the abstract construction is that elements $f\in A$ should be regular functions on the set of maximal ideals in $A$ . Note that by $A$ being a $\mathbb{C}$ -algebra, we have that $\frac A{\mathfrak m} \cong \mathbb{C}$ for any maximal ideal $\mathfrak{m}$ . Therefore, since “ $f(\mathfrak m)$ ” is supposed to be a complex number, a natural choice for this number would be to take $f \pmod{\mathfrak{m}}$ (that is, the residue of $f$ modulo the ideal $\mathfrak m$ will correspond to a number in $\mathbb{C}$ , so take that number). Note that this is informal, since the isomorphism $\frac A{\mathfrak m} \cong \mathbb{C}$ is not canonical.

With this point of view, we can define a Zariski topology on $\mathop{\mathrm{Spec}}_{\mathrm m}(A)$ . Indeed, define the basic open $D(f)$ for $f\in A$ to be the set of maximal ideals $\mathfrak m$ for which $f\neq 0\pmod{\mathfrak m}$ . Then, we can define the sheaf of regular functions on $\mathop{\mathrm{Spec}}_{\mathrm m}(A)$ again by taking $\mathcal{O}_{ \mathop{\mathrm{Spec}}_{\mathrm m}(A) }(D(f)) := A[f^{-1}]$ . It then turns out that if you write $A \cong \frac{\mathbb{C}[x_1,\dots,x_n]}{\sqrt{I}}$ , then we get an isomorphism of (abstract) affine varieties $\mathop{\mathrm{Spec}}_{\mathrm m}(A) \cong V(I)$ .

Example 4: The reason we write $\mathbb{A}^1_{\mathbb{C}}$ for the complex plane viewed as an affine variety is to differentiate it from the affine variety whose coordinate ring is given by $\mathbb{C}$ . Indeed, $\mathop{\mathrm{Spec}}_{\mathrm m}(\mathbb{C})$ is actually given by a singleton point (because $\mathbb{C}$ only has the unique maximal ideal $(0)$ ). On the other hand, we find that $\mathbb{A}^1_{\mathbb{C}} = \mathop{\mathrm{Spec}}_{\mathrm m}(\mathbb{C}[x])$ , where a point $a\in\mathbb{A}^1_{\mathbb{C}}$ corresponds to the maximal ideal $(x-a)$ in $\mathbb{C}[x]$ .

Now we can finally talk about making $\mathop{\mathrm{Spec}}(R)$ look like something geometric for a general ring $R$ . First, to describe the points of this space, you might think we would take the maximal ideals like in the case for affine varieties, but this is wrong. The reason is simply because maximal ideals are hard to retain under general ring homomorphisms. If you check in the case for varieties, a homomorphism of finitely-generated reduced $\mathbb{C}$ -algebras $\varphi:A\to B$ induces a continuous function of maximal spectra $\mathop{\mathrm{Spec}}_{\mathrm m}(B)\to \mathop{\mathrm{Spec}}_{\mathrm m}(A)$ (in the other direction!) by sending a maximal ideal $\mathfrak m$ of $B$ to $\varphi^{-1}(\mathfrak m)$ in $A$ (which turns out to be maximal under these assumptions). However, given a general ring homomorphism $\varphi:R\to S$ and a maximal ideal $\mathfrak m$ of $S$ , the preimage $\varphi^{-1}(\mathfrak m)$ is only guaranteed to be prime.

Fortunately, if $\varphi:R\to S$ and $\mathfrak p$ is prime in $S$ , then $\varphi^{-1}(\mathfrak p)$ will still be prime in $R$ , so we take the points of $\mathop{\mathrm{Spec}}(R)$ to be the prime ideals of $R$ (to relate this to the primitive spectrum of a C*-algebra, this is equivalently the set of kernels of ring homomorphisms into fields). We can then define the Zariski topology on $\mathop{\mathrm{Spec}}(R)$ in an entirely analogous way: define the basic opens $D(f)$ for $f\in R$ to be those primes $\mathfrak p$ for which $f\neq0\pmod{\mathfrak p}$ . Again, the idea is to think of the residue of a ring element $f\in R$ modulo a prime $\mathfrak p$ as the “value” of $f(\mathfrak p)$ , so that we may think of $f$ as a regular function. However, be weary that the hypothetical “codomain” of $f$ changes with the input (the value of $f(\mathfrak p)$ lies in $\frac R{\mathfrak p}$ , and these rings are not in general isomorphic—even if we restrict to maximal ideals).

In accordance with this (albeit slightly awkward) interpretation of ring elements as regular functions, we can equip $\mathop{\mathrm{Spec}}(R)$ with a sheaf of regular functions exactly as before: for $f\in R$ , just set $\mathcal{O}_{ \mathop{\mathrm{Spec}}(R)}(D(f)) := R[f^{-1}]$ , allowing $f$ to be “invertible” where it does not vanish. With this structure, ring homomorphisms $R\to S$ correspond to morphisms of affine schemes $\varphi: \mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ , which are continuous functions equipped with a morphism $\varphi^\sharp : \mathcal{O}_{ \mathop{\mathrm{Spec}}(R)}\to\mathcal{O}_{ \mathop{\mathrm{Spec}}(S)}$ in the other direction. Indeed, the idea is that a ring homomorphism $\psi:R\to S$ induces a map of prime ideals $\varphi: \mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ by setting $\varphi(\mathfrak p) := \psi^{-1}(\mathfrak p)$ , and also induces a morphism of sheaves by acting on $D(f)$ as the induced ring homomorphism $R[f^{-1}]\to S[\psi(f)^{-1}], \frac g{f^k}\mapsto \frac{\psi(g)}{\psi(f)^k}$ . Voilà: we have now made the category $\mathbf{Aff}$ into a category of truly geometric objects!

Side note: If affine varieties weren’t pathological enough as topological spaces, general affine schemes are even stranger: points in $\mathop{\mathrm{Spec}}(R)$ are not even generally closed! (Indeed, the closed points are exactly the ones that correspond to maximal ideals of $R$ .) Moreover, since we are taking arbitrary prime ideals, the spectrum of a finitely-generated reduced $\mathbb{C}$ -algebra will not exactly recover the corresponding affine variety either. For instance, consider $R = \mathbb{C}[x]$ . We still call its affine scheme $\mathbb{A}^1_{\mathbb{C}}$ , but it’s not the same as the affine variety. In particular, we get a point corresponding to the prime ideal $(0)$ , and this point is the furthest from being closed as possible: under the Zariski topology, we find that its closure is the entire space! This means that $(0)$ is “close” to every other point in the space, and so it’s called a generic point of $\mathbb{A}^1_{\mathbb{C}}$ . (Fortunately, you can recover the affine variety by taking the closed points of its corresponding scheme.)