Functorial Algebraic Geometry

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

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