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Unformatted text preview: 1 Some algebraic geometry 1.1 Basic de&amp;nitions. 1.1.1 Zariski and standard topology. Let F be a &amp;eld (almost always R or C ). Then we say that a subset X &amp; F n is Zariski closed if there is a set of polynomials S &amp; F [ x 1 ;:::;x n ] such that X = F n ( S ) = f x 2 F n j s ( x ) = 0 ;s 2 S g . One checks that the set of Zariski closed sets, Z , in F n satis&amp;es the axioms for the closed sets in a topology. That is F n and ; 2 Z . If T &amp; Z then \ Y 2 T Y 2 Z . If Y 1 ;:::;Y m 2 Z then [ j Y j 2 Z . The Hilbert basis theorem (e.g. A.1.2 [GW]) implies that we can take the sets S to be &amp;nite. If X is a subset of F n then the Z-topology on X is the subspace topology corresponding to the Zariski topology. If F = R or C then we can also endow F n with the standard metric topology corresponding to d ( x;y ) = k x y k where k u k 2 = P n i =1 j u i j 2 for u = ( u 1 ;:::;u n ) . We will call this topology the standard topology (in the literature it is also called the classical, Hausdor/or Euclidean topology). Our &amp;rst task will be to study relations be tween these very di/erent topologies. Examples . n = 1 : Then if Y 2 Z then Y = F or Y is &amp;nite. n = 2 : A Zariski closed subset is a &amp;nite union of plane curves or all of F 2 : Thus the Zariski topology on F 2 is not the product topology. 1.1.2 Noetherian topologies. The Zariski topology on a closed subset, X , of F n is an example of a Noetherian topology. That is if Y 1 Y 2 ::: Y m ::: is a decreasing sequence of closed subsets of X then there exists N such that if i;j N then Y i = Y j (this is a direct interpretation of the Hilbert basis theorem). If X is a topological space that be cannot written X = Y [ Z with Y and Z closed and both are proper the X is said to be irreducible. Clearly an irreducible space is connected but the converse is not true. 1 Exercise . What are the irreducible Hausdor/ topological spaces? Lemma 1 Let X be a Noetherian topological space. Then X is a &amp;nite union of irreducible closed subspaces. If X = X 1 [ X 2 [ &amp; &amp; &amp; [ X m with X i closed and irreducible and if X i * X j for i 6 = j then the X i are unique up to order. For a proof see A.1.12 [GW].The decomposition X = X 1 [ X 2 [&amp;&amp;&amp;[ X m with X i closed and irreducible and X i * X j for i 6 = j is called the irredundant decomposition of X into irreducible components. Each of the X i is called an irreducible component . 1.1.3 Nullstellensatz. If X is a subset of F n then we set I X = f f 2 F [ x 1 ;:::;x n ] j f j X = 0 g . If I F [ x 1 ;:::;x n ] is an ideal then we set F n ( I ) = f x 2 F n j f ( x ) = 0 ;f 2 Ig . Lemma 2 A closed subset, X , of F n is irreducible if and only if I X is a prime ideal (i.e. F [ x 1 ;:::;x n ] j X is an integral domain)....
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