81 number of connected components of the corresponding K-set equals the number of distinct roots of F ( Z ) in K . Corollary 9.6. Assume k is a perfect ﬁeld. Let V be a projective algebraic k-set, n = # π0 ( V ) . Then there is an isomorphism of k-algebras O ( V ) ∼ = k 1 ⊕ . . . ⊕ k n where each k i is a ﬁnite ﬁeld extension of k . Moreover n X i [ k i : k ] = #¯ π0 ( V ) . In particular, if V is connected as an algebraic K-set, O ( V ) = K . Proof. Let V 1 , . . . , V n be connected components of V . It is clear that O ( V ) ∼ = O ( V 1 ) ⊕ . . . ⊕O ( V n ) so we may assume that V is connected. Let f ∈ O ( V ) . It deﬁnes a regular map f : V → A 1 ( K ) . Composing it with the inclusion A 1 ( K ) ± → P 1 k ( K ) , we obtain a regular map f0 : V → P 1 k ( K ) . By Theorem 9.1 , f ( V ) = f0 ( V ) is closed in P 1 k ( K ) . Since f ( V ) ⊂ A 1 k ( K ) , it is a proper closed subset, hence ﬁnite. Since V is connected, f ( V ) must be connected (otherwise the pre-image of a connected component of
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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.