LECTURE 1. SYSTEMS OF ALGEBRAIC EQUATIONS
1. For which elds k do the systems
S = cfw_ i (T1 , . . . , Tn ) = 0i=1,.,n , and S = cfw_
Tji = 0i=1,.,n
dene the same a ne algebraic varieties? Here i (T1 , . . . , Tn ) denotes the eleme
Proof. Let f : X ! Y be a morphism. It denes a homomorphism
fO(X ) : Homk (O(X ), O(X ) ! Homk (O(Y ), O(X ).
The image of the identity homomorphism idO(X ) is a homomorphism : O(Y ) !
O(X ). Let us show that ( ) = f . Let 2 X (K ) = Homk (O(X ), K ).
algebras: O(Y ) = k [T1 ] ! O(X ). Every such homomorphism is determined by
its value at T1 , i.e. by an element of O(X ). This gives us one more interpretation
of the elements of the coordinate algebra O(X ). This time they are morphisms
from X to A1
X ! Y of algebraic varieties denes a map fK : X (K ) = V ! Y (K ) =
W of the algebraic sets. So it is natural to take for the denition of regular
maps of algebraic sets the maps arising in this way. We know that f is given
by a homomorphism of k -algeb
Irreducible algebraic sets and
We know that two a ne algebraic k -sets V and V 0 are isomorphic if and only if
their coordinate algebras O(V ) and O(V 0 ) are isomorphic. Assume that both of
these algebras are integral domains
Morphisms of a ne algebraic
In Lecture 1 we dened two systems of algebraic equations to be equivalent if
they have the same sets of solutions. This is very familiar from the theory of
linear equations. However this notion is too strong
Next we shall show that the set of algebraic k -subsets in K n can be used
to dene a unique topology in K n for which these sets are closed subsets. This
follows from the following:
Proposition 2.7. (i) The intersection \s2S Vs of any family cfw_Vs s2S
Systems of algebraic equations
The main objects of study in algebraic geometry are systems of algebraic equations and their sets of solutions. Let k be a eld and k [T1 , . . . , Tn ] = k [T ] be
the algebra of polynomials in n variables over k .
Example 1.1. 1. The system S = cfw_0 k [T1 , . . . , Tn ] denes an a ne algebraic variety denoted by An . It is called the a ne n-space over k . We have, for
any k -algebra K ,
Sol(cfw_0; K ) = K n .
2. The system 1 = 0 denes the empty a ne algebraic
1, . . . , m(r), be the polynomials from I which have the highest coe cient equal
to ai,r . Next, we consider the union J of the ideals Jr . By multiplying a
polynomial F by a power of Tn we see that Jr Jr+1 . This immediately implies
that the union J i
k as a subeld. Note that A is nitely generated as a k -algebra (because k [T ]
is). Suppose we show that A is an algebraic extension of k . Then we will be
able to extend the inclusion k K to a homomorphism A ! K (since K is
algebraically closed), the c
Proof. Let us prove the rst part. If V is irreducible, then the assertion is obvious.
Otherwise, V = V1 [ V2 , where Vi are proper closed subsets of V . If both of
them are irreducible, the assertion is true. Otherwise, one of them, say V1 is