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Unformatted text preview: Problem 0.1. (a) Show that a morphism f : X → Y is of finite type if and only if it is locally of finite type and quasi-compact. (b) Conclude from this that f is of finite type if and only if for every open affine subset V = Spec B of Y , f- 1 ( V ) can be covered by a finite number of open affines U j = Spec A j , where each A j is a finitely generated B-algebra. (c) Show also if f is of finite type, then for every open affine subset V = Spec B ⊆ Y , and for every open affine subset U = Spec A ⊆ f- 1 ( V ), A is a finitely generated B-algebra. Proof. (a) ( ⇒ ) If f is of finite type, then by definition it is also lo- cally of finite type. Next, as f is of finite type, let Y be covered by open affines V i such that f- 1 ( V i ) can be covered by finitely many open affines. As open affines are quasi-compact, it follows that their finite union, f- 1 ( V i ), is quasi-compact. Hence f is also quasi-compact....
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This note was uploaded on 05/02/2010 for the course MATH 632 taught by Professor Staff during the Spring '08 term at University of Michigan.
- Spring '08