INTRODUCTION TO ALGEBRAIC GEOMETRY-page20

INTRODUCTION TO - 16 LECTURE 3 MORPHISMS OF AFFINE ALGEBRAIC VARIETIES Note that it suffices to check the previous condition only for generators of

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Unformatted text preview: 16 LECTURE 3. MORPHISMS OF AFFINE ALGEBRAIC VARIETIES Note that it suffices to check the previous condition only for generators of the ideal I ( Y ) , for example for the polynomials defining the system of equations Y . In terms of the polynomials ( P 1 ( T ) , . . . , P m ( T )) satisfying ( 3.5 ), the morphism f : X → Y is given as follows: f K ( a ) = ( P 1 ( a ) , . . . , P m ( a )) ∈ Y ( K ) , ∀ a ∈ X ( K ) . It follows from the definitions that a morphism φ given by polynomials (( P 1 ( T ) , . . . , P m ( T )) satisfying ( 3.5 ) is an isomorphism if and only if there exist polynomials ( Q 1 ( T ) , . . . , Q n ( T )) such that G ( Q 1 ( T ) , . . . , Q n ( T )) ∈ I ( Y ) , ∀ G ∈ I ( X ) , P i ( Q 1 ( T ) , . . . , Q n ( T )) ≡ T i mod I ( Y ) , i = 1 , . . . , m, Q j ( P 1 ( T ) , . . . , P m ( T )) ≡ T j mod I ( X ) , j = 1 , . . . , n. The main problem of (affine) algebraic geometry is to classify affine algebraic varieties up to isomorphism. Of course, this is a hopelessly difficult problem.varieties up to isomorphism....
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This note was uploaded on 01/08/2012 for the course MATH 299 taught by Professor Wei during the Spring '09 term at SUNY Stony Brook.

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