文档介绍：page 1 of Chapter 8
CHAPTER 8 INTRODUCING ALGEBRAIC GEOMETRY
(COMMUTATIVE ALGEBRA 2)
We will develop enough geometry to allow an appreciation of the Hilbert Nullstellensatz,
and look at some techniques mutative algebra that have geometric significance. As
in Chapter 7, unless otherwise specified, all rings will be mutative.
Varieties
Definitions ments We will be working in k[X1,...,Xn], the ring of poly-
nomials in n variables over the field k. (Any application of the Nullstellensatz requires that
k be algebraically closed, but we will not make this assumption until it es necessary.)
The set An = An(k)ofalln-tuples ponents in k is called affine n- is a
set of polynomials in k[X1,...,Xn], then the zero-set of S, that is, the set V = V (S) of all
x ∈ An such that f(x) = 0 for every f ∈ S, is called a variety. (The term “affine variety”
is more precise, but we will use the short form because we will not be discussing projective
varieties.) Thus a variety is the solution set of simultaneous polynomial equations.
If I is the ideal generated by S, then I consists of all finite binations gifi
with gi ∈ k[X1,...,Xn] and fi ∈ S. It follows that V (S)=V (I), so every variety is the
variety of some ideal. We now prove that we can make An into a topological space by taking
varieties as the closed sets.
Proposition
(1) If Vα= V (Iα) for all α∈ T , then Vα= V ( Iα). Thus an arbitrary intersection of
varieties is a variety.
r { ··· ∈≤≤}
(2) If Vj = V (Ij),j =1,...,r, then j=1 Vj = V ( f1 fr : fj Ij, 1 j r ). Thus a
finite union of varieties is a variety.
(3) An = V (0) and ∅= V (1), so the entire space and the empty set are varieties.
Consequently, there is a topology on An, called the Zariski topology, such that the
closed sets and the varieties coincide.
Proof.
n
(1) If x ∈ A , then x ∈ Vα iff every polynomial in every Iα vanishes at x iff x ∈ V ( Iα).
∈ r ∈
(2) x j=1 Vj iff for some j, every fj Ij vanishe