Algebraic geometry is the study of subsets of affine n
and, more generally projective space
, defined by the vanishing
This theory has been an amazingly successful area of mathematics
years. One reason for this
is that the important
geometric spaces, the ones
that turn up in applications (such as
), 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
just empty formalism. For example,
algebraic geometry is everywhere present in
' 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
X corresponds to the ideal I(X).
Its natural then to define the coordinate ring of X
to be the quotient ring
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
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)
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].