Suppose $X$ and $Y$ are separated noetherian schemes. Let $f: X\to Y$ be a morphism between them. If $\mathcal{U}=(U_{i})$ is an open affine cover of $X$ and $V$ is any open affine subset of $Y$, Hartshorne's III proposition 8.7 claims that $U_{i}\cap f^{-1}(V)$ are still open affine subsets of $X$.

Hartshorne gives a hint II Ex 4.3, but $f^{-1}(V)$ may not be an open affine subset of $X$. And I don't know how to use $V$ is an open affine subset of $Y$... And I don't know whether $f$ is separated and I can't use properties of fibre product... Maybe I misunderstand this problem...

Could you help me and give me an explanation?


1 Answer 1


$\require{AMScd}$First, if $S$ is a scheme, and $W, W' \subset S$ are open, then $W \cap W' = W \times_S W'$.

We may also write $f^{-1}(V) = X \times_Y V$, so that $$U \cap f^{-1}(V) = U \times_X (X \times_Y V) \cong (U \times_X X) \times_Y V = U \times_Y V.$$

To finish up, we write the corresponding cartesian diagram below.

\begin{CD} U \times_Y V @>>> V\\ @VVV @VVV\\ U @>>> Y \end{CD}

Now, since $Y$ is separated, the morphism $V \to Y$ is an affine morphism by the mentioned exercise. Hence, as the base change of such a morphism, $U \times_Y V \to U$ is also affine, which implies that $U \times_Y V$ is an affine scheme.


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