 
Summary: Algebraic Geometry I
Lectures 3, 4, and 5
Amod Agashe
October 13, 2008
1 Varieties
Proposition 1.1. (a) If X and Y are algebraic sets in An
, then so is XY
(b) If I is a set and i I, Xi is an algebraic set in An
, then iI Xi is
an algebraic set.
(c) and An
are algebraic sets.
Proof. (a) Let X = Z(T1) and Y = Z(T2), where T1 and T2 are subsets in
k[x1, . . . , xn]. Then X Y = Z(T1T2) = {f1f2f1 T1, f2 T2} whence
X Y is an algebraic set.
(b) Let Xi = Z(Ti) i I. Then iI Xi = Z( iI Ti).
(c) Note that = Z(1) and An
= Z(0).
So the algebraic sets are the closed sets of a topology, called the Zariski
topology. In other words, in the Zariski topology on An
