Chevalley’s Theorem and Complete Varieties
نویسنده
چکیده
Proof. Let f1, . . . , fm be generators of A(X) over A(Y ). Then the fi also generate K(X) over K(Y ), so we can reorder indices such that f1, . . . , fr are algebraically independent over K(Y ). Let R = A(Y )[f1, . . . , fr] ⊆ A(X). Since the fi are algebraically independent over K(Y ) they are algebraically independent over A(Y ), so R is isomorphic to an r-variable polynomial ring, which is to say that R ∼= A(Y × Ar). Then the inclusions A(Y ) ↪→ R ↪→ A(X) induce the desired factorization.
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