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  "body": "\n## Definition\n\nA [[morphism of schemes|JXRKH6P1Z4]] $f \\colon X \\to Y$ is *finite* if there is a cover of $Y$ by open [[affine|B3V3XJ2W8G]] subschemes $V_i = \\operatorname{Spec} B_i$ such that, for each $i$:\n\n1. $f^{-1}(V_i)$ is an affine scheme $\\operatorname{Spec} A_i$;\n2. $A_i$, made into a $B_i$-module by the ring map $B_i \\to A_i$ of $f$, is a [[finitely generated module|05QY1VEBYD]].\n\n## Remarks\n\nIf $f$ is finite, then clauses 1 and 2 hold for every open affine $V \\subseteq Y$.\nA morphism $\\operatorname{Spec} A \\to \\operatorname{Spec} B$ is finite if and only if $B \\to A$ is a [[finite morphism of rings|1NBACAEC2V]].\nA finite morphism is [[affine|AFC6JMYWEH]] and of finite type.\nExample: for a field $k$, the morphism $\\mathbb{A}^1_k \\to \\mathbb{A}^1_k$ given by $k[x] \\to k[t]$, $x \\mapsto t^2$, is finite, since $k[t] = k[x] \\cdot 1 + k[x] \\cdot t$.\nNon-example: the open immersion $\\operatorname{Spec} k[x, x^{-1}] \\to \\operatorname{Spec} k[x]$ satisfies clause 1 but not clause 2.\nIf $k[x, x^{-1}]$ were a finite $k[x]$-module, then $x^{-1}$ would be integral over $k[x]$.\nSince $k[x]$ is an [[integrally closed domain|H20P600HQS]], this would give $x^{-1} \\in k[x]$, which is false.\n",
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