Hartshorne2.3.3 - Problem 0.1(a Show that a morphism f X Y...

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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. ( ) Take a cover of Y by V i ’s from f being locally of finite type, so that f - 1 ( V i ) is covered by open affines that give finitely generated algebras.
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