I'm going to move on with the identification of geometric objects with
functions from these objects down to a field soon enough, but I'd like
to spend a little time nailing down the categorical language of this
association. Basically, we have two functors I and V going back and
forth between two categories. And the essential statement of the last
post is that these two functors form an equivalence of categories.

Now, first off in this categorical language, I want to nail down exactly
what the objects are. In the category [tex]\mathcal{AV}ar_k[/tex] the
objects are solution sets of systems of polynomial equations. And in the
category [tex]\mathcal{RA}lg_k[/tex], the objects are finitely
presented Noetherian reduced k-algebras.

The functor [tex]V:\mathcal{RA}lg_k\to \mathcal{AV}ar_k[/tex] acts
on objects by sending an algebra R to the solution set of the polynomial
equations generating the ideal in a presentation of the algebra.

And the functor [tex]I:\mathcal{AV}ar_k\to \mathcal{RA}lg_k[/tex]
takes a variety to the algebra of polynomial functions on the variety.
This is a slight modification to the way I've introduced I in the
previous posts - but the good news is that I(V) is the quotient of the
right polynomial ring with the previously defined I(V).

## Morphisms on varieties

In order to define a category, it's not enough with the objects - we
want morphisms as well. Since everything else is defined by polynomials,
we're going to define a morphism of varieties [tex]V\to W[/tex] to be a
map [tex]\mathbb A^n\to\mathbb A^m[/tex], polynomial in each
coordinate, such that the image of the restriction to the variety
[tex]V\subseteq\mathbb A^n[/tex] is contained in
[tex]W\subseteq\mathbb A^m[/tex].

In other words, it is a map
[tex](x_1,\dots,x_n)\mapsto(f_1(x_1,\dots,x_n),\dots,f_m(x_1,\dots,x_n))[/tex]
defined by polynomials [tex]f_1,\dots,f_m[/tex].

An isomorphism of varieties is exactly what we expect it to be - it is a
morphism that has an inverse.

One specific kind of highly interesting isomorphisms are the linear
automorphisms of [tex]\mathbb A^n[/tex]. These are given, essentially,
by invertible change-of-coordinate matrices in the way we are used to
from linear algebra.

### Examples

Recall from the earlier posts the twisted cubic curve
[tex]V(x-y^2,x-z^3)[/tex]. Points on it have the form
[tex](s,s^2,s^3)[/tex] - and this, incidentially, gives us precisely a
map [tex]\mathbb A^1\to\mathbb A^3[/tex] displaying an isomorphism
between the twisted cubic curve and the affine line. The inverse is
given by [tex](x,y,z)\mapsto x[/tex].

Consider the parabola [tex]V(x-y^2)[/tex]. This is also isomorphic to
the affine line, over the maps [tex]t\mapsto(t,t^2)[/tex] and
[tex](x,y)\mapsto x[/tex].

On the other hand, the affine line is not isomorphic to the nodal curve
[tex]V(y^2-x^2-x^3)[/tex]. The easiest way to show this is to go over
smoothness of curves and singular points - which I hope to deal with
later at some point. Essentially, smoothness is an invariant of
varieties under isomorphisms, and since the point (0,0) is singular on
the nodal curve, and the affine line has no singular points, the two
varieties can not possibly be isomorphic.

Note that images of varieties need not be affine algebraic varieties -
they will, however, always be *quasi-projective* varieties. We'll see if
I get into this later on.

## Morphisms of algebras and functoriality of V and I

We really do already know what morphisms look like in
[tex]\mathcal{RA}lg_k[/tex]. This category is the full subcategory of
the category of k-algebras - by which we mean that it picks out objects
among k-algebras, and have all k-algebra maps between objects as
morphisms.

The really awesome bit happens when we start considering the morphisms
we've defined. Given a morphism [tex]F:V\to W[/tex], we define the
pullback [tex]F^\#:k[W]\to k[V][/tex] by [tex]F^\#(f)=f\circ
F[/tex]. This takes a map [tex]f:W\to k[/tex] and makes a map
[tex]F^\#(f):V\to k[/tex]. Since this is a composition of polynomials,
it is also a polynomial function. If [tex]f\in I(V)[/tex], then
[tex]F^\#(f)(p)=f(F(p))[/tex], and since [tex]F(p)\in V[/tex], it
follows that [tex]f(F(p))=0[/tex], and thus [tex]F^\#(f)\in
I(W)[/tex].

In the other direction, suppose that R and S are reduced finitely
generated k-algebras. Then [tex]R=k[x_1,\dots,x_r]/I[/tex], and
[tex]S=k[y_1,\dots,y_s]/J[/tex]. We fix a homomorphism
[tex]\sigma:R\to S[/tex], and we wish to construct a variety morphism
[tex]F[/tex] such that [tex]F^\#=\sigma[/tex].

Let [tex]F_i\in k[y_1,\dots,y_s][/tex] be a representative of
[tex]\sigma(x_i)[/tex], and define [tex]F:\mathbb A^s\to \mathbb
A^r[/tex] by [tex]a\mapsto(F_1(a),\dots,F_r(a))[/tex]. We need to
verify that F maps V to W. This follows if we can only show that for
every [tex]a\in V[/tex], all polynomials in I vanish on F(a). Let
[tex]g\in I[/tex]. Then

[tex]g(F(a))=g(F_1(a),\dots,F_r(a))=g(\sigma(x_1)(a),\dots,\sigma(x_r)(a))=\sigma(g)(a)[/tex]

and since [tex]g\in I[/tex], it represents the zero class of
[tex]k[x_1,\dots,x_r]/I[/tex], so [tex]\sigma(g)=0[/tex] and hence
[tex]\sigma(g)\in J[/tex]. But J is the ideal of all functions
vanishing on V. Hence [tex]F(a)\in W[/tex], and the proof is
complete.

In essence, what this proves to us is that the operations V and I form a
contravariant equivalence of categories between
[tex]V:\mathcal{RA}lg_k[/tex] and [tex]\mathcal{AV}ar_k[/tex].