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].

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