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 to0 . 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 7→ 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
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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.