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Let

  • $f: X\to Y$ be a quasi compact separated morphism

  • $\{U_i\}$ be an open affine covering of $X$

  • $V$ be an open affine subset of $Y$

In Hartshorne's AG book chapter III, proposition 8.7 uses ;

$\{U_i \cap f^{-1}(V)\}$ forms an open affine cover of $f^{-1}(V)$

Is it really true?

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Thanks for your kind edit, David! –  choa Oct 31 '11 at 6:38
    
You're welcome! –  David Roberts Oct 31 '11 at 6:44
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1 Answer

up vote 1 down vote accepted

No, this is not true, but it is not what Hartshorne uses. You need $Y$ separated, then the cartesian square $$\begin{matrix} U_i\cap f^{-1}(V) & \to & U_i\times V \\\\ \downarrow && \downarrow \\\\ Y & \to & Y\times Y \end{matrix}$$ exhibits $U_i\cap f^{-1}(V)$ as a closed subscheme of the affine scheme $U_i\times V$. See also EGA I (Springer edition), 5.3.10.

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Thanks! the cartesian diagram makes it clear. –  choa Oct 31 '11 at 6:37
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