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INTRODUCTION TO ALGEBRAIC GEOMETRY-page20

INTRODUCTION TO ALGEBRAIC GEOMETRY-page20 - 16 LECTURE 3...

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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. Example 3.3. 1. Let Y be given by the equation T 2 1 - T 3 2 = 0 , and X = A 1 k with O ( X ) = k [ T ] . A morphism f : X Y is given by the pair of polynomials ( T 3 , T 2 ) . For every k -algebra K , f K ( a ) = ( a 3 , a 2 ) Y ( K ) , a X ( K ) = K. The affine algebraic varieties X and Y are not isomorphic since their coor- dinate rings are not isomorphic. The quotient field of the algebra
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