Math 697R

Applicable Algebraic Geometry

Day 9: Thurdsay, 12 Ocotber

* Wednesday 25 October ~5--6:30   Make up session
* Sunday, 15 October 5:30 PM  89 N Prospect Street
* Please feel free to email me, mentioning any notions I 
   used that were not immediately clear to you---particularly
   notions that I treated as background that I assumed.
* The notes on polynomial matrices are one rewrite away from distribution

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 - Theorem:  Any affine variety is a finite union of irreducible 
           affine varieties
 - Fact: Irreducible Complex varieties are connected in the usual topology
   ! not true for Real varieties
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Regular and Rational Functions

 - Def: Regular function
        Can add & multiply; they form a K-algebra
        map K[A^n] -->> K[X] has kernel I(X)
 -  Sometimes when K is not closed & X=V(I) with I radical, let
        K[X] = K[A^n]/I
 -  Coordinate ring is a finitely generated algebra w/o nilpotents
           (reduced)
 - Theorem:  That property characterizes coordinate rings of 
             affine algebraic varieties, when K is alg. closed
 --> Different generators give different affine varieties !
 -  Subvarieties of X and ideal of K[X].
 - Hilbert Theiorems for  X, K[X].
 - Algebraic - Geometric Dictionary for X, K[X]
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 - Def: Map given by list of regular functions.
        Pullback & its kernel/image
 - Ring map gives map of varieties
 - Example: Matrix Multiplication
 - Definition: Isomorphism
 - Example:  t |--> (t, t^2)  is isomorphism
             t |--> (t^2, t^3) is bijection, not isomorphism

 - Algebraic - Geometric Dictionary as a category equvalence
 - Principal affine sets
 - K^x & GLn