AG studies the zeros of polynomial functions, but what are polynomial functions?

Algebraic Geometry Studies Zeros of Polynomial Functions. But What Is a Polynomial Function?A final Lawvere-theoretic reformulation

 

Algebraic Geometry Studies Zeros of Polynomial Functions. But What Is a Polynomial Function?

A standard slogan says that algebraic geometry studies zero loci of polynomial functions. But this immediately raises a basic question:

What is a polynomial function?

If we fix one ring $R$$R$, a polynomial

$$f(x_1,\dots ,x_n)\in \mathbb{Z}[x_1,\dots ,x_n]$$

certainly defines a function

$$R^n\to R$$

by evaluation. But this is not the right concept. Over a finite field, different polynomials may induce the same set-theoretic function. So a polynomial should not be identified with a function on one fixed set of points.

The correct viewpoint is functorial.

Let

$$P_n=\mathbb{Z}[x_1,\dots ,x_n].$$

For every commutative ring $R$$R$, the universal property of the polynomial ring gives a natural bijection

$${\mathrm{Hom}}_{\mathbf{CRing}}(P_n,R)\cong R^n.$$

Thus affine $n$$n$-space is better viewed as the functor

$$h_{P_n}:\mathbf{CRing}\to \mathbf{Set},h_{P_n}(R)=\mathrm{Hom}(P_n,R).$$

Now suppose we want a polynomial map from affine $n$$n$-space to affine $m$$m$-space. Functorially, this should be a natural transformation

$$\varphi :h_{P_n}\to h_{P_m}.$$

The Yoneda Lemma gives

$$\mathrm{Nat}(h_{P_n},h_{P_m})\cong h_{P_m}(P_n)=\mathrm{Hom}(P_m,P_n).$$

So polynomial maps are exactly natural transformations between these functors.

Let us write out the naturality square in the usual style, with the natural transformation vertical. For any ring map

$$p:P_n\to R,$$

we have a commutative diagram

$$\begin{array}{ccc}{h}_{P_n}(P_n)& \stackrel{h_{P_n}(p)}{\to }& h_{P_n}(R)\\ \downarrow {\varphi }_{P_n}& & \downarrow {\varphi }_R\\ h_{P_m}(P_n)& \stackrel{h_{P_m}(p)}{\to }& h_{P_m}(R).\end{array}$$

Now let

$$u={\varphi }_{P_n}({\mathrm{id}}_{P_n})\in h_{P_m}(P_n)=\mathrm{Hom}(P_m,P_n).$$

Evaluating the diagram at ${\mathrm{id}}_{P_n}$$\operatorname{id}_{P_n}$ gives

$${\varphi }_R(p)=h_{P_m}(p)(u)=p\circ u.$$

If we write

$$P_m=\mathbb{Z}[y_1,\dots ,y_m],$$

then a ring map

$$u:P_m\to P_n$$

is determined by the images of the generators:

$$u(y_j)=f_j(x_1,\dots ,x_n).$$

Hence a natural transformation

$$h_{P_n}\to h_{P_m}$$

is exactly the same thing as an $m$$m$-tuple of polynomials

$$(f_1,\dots ,f_m).$$

Under the identification $h_{P_n}(R)\cong R^n$$h_{P_n}(R)\cong R^n$, the induced map is

$$(a_1,\dots ,a_n)\mapsto (f_1(a),\dots ,f_m(a)).$$

This is why polynomial functions should be defined as natural transformations.

Now consider a system of equations

$$f_1=\cdots =f_m=0.$$

It determines a map

$$u:P_m\to P_n,y_j\mapsto f_j,$$

and also the zero map

$$z:P_m\to P_n,y_j\mapsto 0.$$

These induce two natural transformations

$$h_{P_n}\rightrightarrows h_{P_m}.$$

Their equalizer is the solution functor:

$${\mathrm{Sol}}_f(R)=\{p:P_n\to R\mid p\circ u=p\circ z\}.$$

Since it is equalizer of representable functor, it is represent by coequalizer.

$${\mathrm{Sol}}_f\cong h_{P_n/(f_1,\dots ,f_m)}.$$

This is the basic affine dictionary:

$$\text{equations}⟷\text{quotient rings}⟷\text{solution functors}.$$

Finally, if we allow an arbitrary set $I$$I$ of variables, we may form

$$P_I=\mathbb{Z}[x_i\mid i\in I].$$

Then

$$\mathrm{Hom}(P_I,R)\cong R^I,$$

so everything above still works. Moreover, every commutative ring $A$$A$ admits a presentation

$$A\cong \mathbb{Z}[x_i\mid i\in I]/(f_j\mid j\in J)$$

for suitable sets $I$$I$ and $J$$J$. Hence

$$h_A(R)=\mathrm{Hom}(A,R)\cong \{(a_i)\in R^I\mid f_j(a_i)=0\text{ for all }j\in J\}.$$

So every affine scheme is a functor of solutions to a system of polynomial equations.

This refines the opening slogan. Algebraic geometry does study zeros of polynomial functions — but polynomial functions are not merely set-theoretic functions on one fixed ring. They are natural transformations between affine-space functors, and affine schemes are precisely the representable solution functors cut out by such maps.

A final Lawvere-theoretic reformulation

There is one last way to summarize the whole discussion.

Let

$$U:\mathrm{CRing}\to \mathrm{Set}$$

be the forgetful functor. Then affine $n$$n$-space is simply

$${\mathbb{A}}^n=U^n.$$

The natural transformations

$$U^n\Rightarrow U^m$$

are exactly the polynomial maps

$${\mathbb{A}}^n\to {\mathbb{A}}^m.$$

Therefore, if we take the category whose objects are

$$U^0,U^1,U^2,\dots$$

and whose morphisms are the natural transformations between them, we recover precisely the Lawvere theory of commutative rings.

So the whole point can be compressed into the slogan:

$$\text{finite affine spaces and polynomial maps}=\text{the Lawvere theory of commutative rings}.$$

This is another way to say that the syntax of commutative algebra is encoded in the natural geometry of the functors

$$A\mapsto A^n.$$

For more on this structure-versus-semantics viewpoint, see my related post:

Structure–Semantics Adjunction for Tractable Functors