lectures-ch-1-math207

lectures-ch-1-math207 - 1 Some algebraic geometry 1.1 Basic

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 Some algebraic geometry 1.1 Basic de&nitions. 1.1.1 Zariski and standard topology. Let F be a &eld (almost always R or C ). Then we say that a subset X & F n is Zariski closed if there is a set of polynomials S & 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&es the axioms for the closed sets in a topology. That is F n and ; 2 Z . If T & 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 &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 &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 &nite. n = 2 : A Zariski closed subset is a &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 &nite union of irreducible closed subspaces. If X = X 1 [ X 2 [ & & & [ 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 [&&&[ 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)....
View Full Document

This note was uploaded on 02/28/2012 for the course MATH 207C taught by Professor N.r.wallach during the Spring '10 term at Colorado.

Page1 / 20

lectures-ch-1-math207 - 1 Some algebraic geometry 1.1 Basic

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online