Affine morphism

Remarks

If $f$ is affine, then $f^{-1}(V)$ is affine for every open affine subscheme $V$ of $Y$. Example: every morphism $\operatorname{Spec} A \to \operatorname{Spec} B$ is affine; take the cover of $Y = \operatorname{Spec} B$ by itself, with preimage $\operatorname{Spec} A$. Example: a closed immersion is affine, since over $\operatorname{Spec} B$ it is $\operatorname{Spec} B/I \to \operatorname{Spec} B$ for an ideal $I$. Non-example: for a field $k$, the morphism $\mathbb{A}^2_k \setminus \set{0} \to \operatorname{Spec} k$ is not affine. The only open affine of $\operatorname{Spec} k$ is the whole space, and its preimage $\mathbb{A}^2_k \setminus \set{0}$ is not an affine scheme. Its ring of global functions is $k[x, y]$, whose spectrum also contains the origin.