Inverse image of affine open set on separated scheme

algebraic-geometry

Soory for my bad English.

Let $X, Y$ be Noetherian separated scheme, and $f:X\to Y$ be morphism of scheme.

Let $U \subseteq Y$ be affine open subscheme.

Then, Is $f^{-1}(U)=X\times_Y U$ affine?

Tell me proof or counter example ,thanks.

Best Answer

No, morphisms with that property are called affine, see e.g. stacks project. For a counterexample, just pick a non-affine Noetherian separated scheme over a field $k$ and consider the structure morphism to $\text{Spec}(k)$.

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