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  "body": "\n## Definition\n\nA [[morphism of schemes|JXRKH6P1Z4]] $f \\colon X \\to Y$ is *affine* if $Y$ has a cover by open [[affine|B3V3XJ2W8G]] subschemes $V_i$ with each $f^{-1}(V_i)$ affine.\n\n## Remarks\n\nIf $f$ is affine, then $f^{-1}(V)$ is affine for every open affine subscheme $V$ of $Y$.\nExample: every morphism $\\operatorname{Spec} A \\to \\operatorname{Spec} B$ is affine; take the cover of $Y = \\operatorname{Spec} B$ by itself, with preimage $\\operatorname{Spec} A$.\nExample: a closed immersion is affine, since over $\\operatorname{Spec} B$ it is $\\operatorname{Spec} B/I \\to \\operatorname{Spec} B$ for an ideal $I$.\nNon-example: for a field $k$, the morphism $\\mathbb{A}^2_k \\setminus \\set{0} \\to \\operatorname{Spec} k$ is not affine.\nThe 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.\nIts ring of global functions is $k[x, y]$, whose spectrum also contains the origin.\n",
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