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
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