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  "body": "\n## Definition\n\nLet $f \\colon X \\to Y$ be a [[morphism of schemes|JXRKH6P1Z4]].\nIt is *locally of finite type* if there is a cover of $Y$ by open [[affine|B3V3XJ2W8G]] subschemes $V_i = \\operatorname{Spec} B_i$ with the following property.\nFor each $i$, the open set $f^{-1}(V_i)$ is covered by open affine subschemes $U_{ij} = \\operatorname{Spec} A_{ij}$, where each $A_{ij}$ is a [[finitely generated|AC5TFFQT9M]] $B_i$-algebra.\nIt is *of finite type* if moreover each $f^{-1}(V_i)$ can be covered by finitely many such $U_{ij}$.\n\n## Remarks\n\nEquivalently, $f$ is of finite type if and only if it is locally of finite type and [[quasi-compact|41009J4M5D]].\nA morphism $\\operatorname{Spec} A \\to \\operatorname{Spec} B$ is of finite type if and only if $A$ is a finitely generated $B$-algebra.\nExample: for a field $k$, the morphism $\\mathbb{A}^n_k \\to \\operatorname{Spec} k$ is of finite type, since $k[x_1, \\ldots, x_n]$ is generated by the $x_i$.\nProjective space $\\mathbb{P}^n_k \\to \\operatorname{Spec} k$ is of finite type too, being covered by $n + 1$ such affine pieces.\nNon-example: $\\operatorname{Spec} k(x) \\to \\operatorname{Spec} k$ is not of finite type.\nIf the field $k(x)$ were a finitely generated $k$-algebra, then by [[Zariski's lemma|YBJ2VQHFDQ]] it would be a finite extension of $k$, which it is not.\n",
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