Algebraic geometry is the study of subsets of affine
n-space
and, more generally
projective space, defined by the vanishing
of
polynomials.
This theory has been an amazingly successful area of
mathematics
in recent
years. One reason for this
is that the
important geometric spaces, the ones
that turn up in applications (such as
theoretical physics), are nearly
always algebraic and so can be attacked by the methods of algebraic geometry.
What's more the techniques used in algebraic geometry have become
ubiquitous in other areas of
mathematics. In the sixties
Alexander Grothendieck
generalised the whole thing massively
when he introduced the idea of a
scheme. Once one has this notion then
all of algebraic
number theory becomes part of algebraic geometry. This
is
not just empty formalism. For example,
algebraic geometry is everywhere present in
Andrew Wiles' proof of
Fermat's Last Theorem (of course he was trying
to prove a
conjecture
about
elliptic curves).
Nowadays there is even
noncommutative algebraic geometry!
Let's be a bit more specific and talk about one of the basic results in
algebraic geometry. I won't talk about schemes, let's save that for
another day. So fix an algebraically closed base field k.
Recall from the Zariski topology writeup the definition of
an affine variety. Let X inside kn be such
a variety. So X is a closed subset in the Zariski toplology.
There is a bijective correspondence between such varieties
and radical ideals in
k[x1,...,xn]. Here
X corresponds to the ideal I(X).
Its natural then to define the coordinate ring of X
to be the quotient ring
k[X]=k[x1,...,xn]/I(X).
Note that if one thinks of the polynomials in
k[x1,...,xn] as functions on X
then the ones acting as the zero function are exactly the elements of
I(X). This means that the elements of k[X]
can be identifed with functions on X.
As usual we now want to have maps between our varieties.
Let X <= kn and Y <= km
be two varieties. A polynomial map f:X-->Y is a function
such that there exist m elements
f1,...,fm of
k[X]
such that
f(x)=(f1(x),...,fm(x))
for all x in X.
Note that whenever we have such a polynomial map we can define
a ring homomorphism f*:k[Y]-->k[X]
by the mapping f*(h)=hf, where the latter is
interpreted as a function on X.
Since we now have objects, namely affine varieties, and morphisms,
namely polynomial maps, then we have a category of affine varieties.
A result that is crucial pyschologically,
tells us that we can freely switch between varieties
and rings. A commutative ring is called a k-algebra if it
has k as a subring. It is called finitely generated if there
is a surjective ring homomorphism (that is the identity on k)
from a polynomial ring (in finitely many variables)
to it.
Theorem
There is an arrow-reversing equivalence of categories
between the category of varieties and the category of finitely generated
reduced commutative k-algebras. The equivalence is given by mapping
a variety X to its coordinate ring k[X]
and mapping a polynomial map f:X-->Y to
the ring homomorphism f*:k[Y]-->k[X].