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morphism of finite type

DEFINITION 10.1   Let $ X$ , $ Y$ be schemes, and let $ f:X\to Y$ be a morphism. We say that $ f$ is of finite type if there exists an open cover $ \{U_i\}$ of $ Y$ by affine schemes and a finite open cover $ \{V_{i j}\}$ of each $ f^{-1}(U_i)$ by affine schemes such that $ f_{i j}= f\vert _{V_{i j}}$ is ``a morphism of finite type'' for every $ i$ and $ j$ . That means, if we put

$\displaystyle \Gamma_{f_{i j}}:
A_i=\Gamma(\mathcal{O}_{U_i})
\to \Gamma(\mathcal{O}_{V_{i j}}) =B_{i j}
$

Then $ B_{i j}$ is finitely generated algebra over $ A_{i}$ .



2007-12-11