INTRODUCTION TO ALGEBRAIC GEOMETRY-page91

INTRODUCTION TO ALGEBRAIC GEOMETRY-page91 - 87 Example 10.4...

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87 Example 10.4. 1. Let X = { ( x, y ) K 2 : y = x 2 } ⊂ A 2 ( K ) and Y = A 1 ( K ) . Consider the projection map f : X Y, ( x, y ) y . Then f is finite. Indeed, O ( X ) = k [ Z 1 , Z 2 ] / ( Z 2 - Z 2 1 ) , O ( Y ) = k [ Z 2 ] and f * is the composition of the natural inclusion k [ Z 2 ] k [ Z 1 , Z 2 ] and the natural homomorphism k [ Z 1 , Z 2 ] k [ Z 1 , Z 2 ] / ( Z 2 - Z 2 1 ) . Obviously, it is injective. Let z 1 , z 2 be the images of Z 1 and Z 2 in the factor ring k [ Z 1 , Z 2 ] / ( Z 2 - Z 2 1 ) . Then O ( X ) is generated over f * ( O ( Y )) by one element z 1 . The latter satisfies a monic equation: z 2 1 - f * ( Z 2 ) = 0 with coefficients in f * ( O ( Y )) . As we saw in the proof of Lemma 10.1 , this implies that O ( X ) is a finitely generated f * ( O ( Y )) -module and hence O ( X ) is integral over f * ( O ( Y )) . Therefore f is a finite map. 2. Let x 0 be a projective subspace of P n k ( K ) of dimension 0 , i.e., a point ( a 0 , . . . , a n ) with coordinates in k . Let X be a projective algebraic k -set in P n k ( K ) with x 0 X and let f = pr x 0 : X P n - 1 k ( K ) be the projection map. We know that Y = f ( X ) is a projective set. Let us see that f : X Y is finite. First, by a variable change,
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