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INTRODUCTION TO ALGEBRAIC GEOMETRY-page30

# INTRODUCTION TO ALGEBRAIC GEOMETRY-page30 - 26LECTURE 4...

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26 LECTURE 4. IRREDUCIBLE ALGEBRAIC SETS AND RATIONAL FUNCTIONS We have P ( T 1 , . . . , T n ) 0 on V \ Z . Since V \ Z is dense in the Zariski topology, P 0 on V , i.e. P I ( V ) . This shows that under the map R ( W ) R ( V ) , F goes to 0 . Since the homomorphism R ( W ) R ( V ) is injective (any homomorphism of fields is injective) this is absurd. In particular, taking W = A 1 k ( K ) , we obtain the interpretation of elements of the field R ( V ) as non-constant rational functions V - K defined on an open subset of V (the complement of the set of the zeroes of the denominator). From this point of view, the homomorphism R ( W ) R ( V ) defining a rational map f : V - → W can be interpreted as the homomorphism f * defined by the composition φ φ f . Definition 4.4. A rational map f : V - → W is called birational if the cor- responding field homomorphism f * : R ( W ) R ( V ) is an isomorphism. Two irreducible affine algebraic sets V and W are said to be birationally isomorphic if there exists a birational map from V to W .
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