Lecture 1

Basic Notions

This lecture provides a summary of basic notions concerning algebraic sets,

i.e., solution sets of polynomial equations. We will discuss the notion of

dimension of an algebraic set and review the required results from commu-

tative algebra along the way.

Throughout this lecture, K will denote a field. The rings considered will

be commutative and with an identity element.

1. Algebraic sets

Definition 1.1. Let R = K[xi,..., xn] be a polynomial ring in n variables

over a field K, and consider polynomials / i , . . . , fm G R. Their zero set

{(ai,... , an) G

Kn

| /i(ai,... , an) = 0 for 1 ^ i m}

is an algebraic set in

Kn,

denoted Var(/i,..., f

m

) . These are our basic

objects of study, and they include many familiar examples.

Example 1.2. If / i , . . . , fm G K[x\,..., xn] are homogeneous linear poly-

nomials, their zero set is a vector subspace of

Kn.

If V and W are vector

subspaces of

Kn,

then we have the following inequality:

rankK(V^ n W) rankK V + rankjc W — n.

One way to prove this inequality is by using the exact sequence

o • vnw

—?-*

v®w

-^-^ v + w

• o

where a(u) = (u, u) and j3(v, w) = v — w. Then

rankK(V n W) =

rankK(Vr

®W) - rankK(V + W)

^ rankK V + rankjK W — n.

T

http://dx.doi.org/10.1090/gsm/087/01