Algebraic geometry

Algebraic geometry is the study of the solutions of algebraic equations. Let F be a field, and let S be a set of polynomials in n variables x1, ... , xn. The set V=V(S) of all points a=(a1,...,an) in F^n which are solutions to all f in S is called an affine variety. Conversely if I=I(V) is the set of all polynomials vanishing on V, then I is an ideal in the polynomial ring F[x1,...,xn] (ring ideal). The quotient ring A=F[x1,...,xn]/I is called the coordinate ring of the affine variety V. This correspondence between affine varieties and their coordinate rings is particularly clear when the ground field F is algebraically closed, due to a theorem of Hilbert's called the Nullstellensatz. It says that the points of V are in one to one correspondence with the maximal ideals of A. Because of this relationship the fields of algebraic geometry and commutative algebra (i.e. commutative ring theory) are very closely related.

Affine varieties are special cases of more general objects called algebraic varieties. A special case is that of a projective variety. This is a set in projective space whose intersection with any affine subspace is an affine variety. Varieties are given an important topology called the Zariski topology in which the closed subsets are the subvarieties.

Algebraic geometry was developed largely by the Italian geometers in the early part of the 20-th century. Their work was deep but not on a sufficiently rigorous basis. Commutative algebra was developed by Hilbert, Emmy Noether and others, also in the 20-th century, with the geometric applications in mind. In the 1930's and 1940's Andre Weil realized that putting algebraic geometry on a rigorous basis was needed and he gave such a theory. In the 1950's and 1960's Serre and particularly Grothendieck recast the foundations making use of the theory of sheaves. In Grothendieck's formulation the study of algebraic varieties has been replaced by that of more abstract objects called schemes.