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?



Question about affine open covering
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.


