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  "body": "\n## Definition\n\nA [[scheme|5A35TYZD02]] $X$ is *locally Noetherian* if it can be covered by open [[affine schemes|B3V3XJ2W8G]] $\\operatorname{Spec} A_i$ in which every $A_i$ is a [[Noetherian ring|0670T59HGE]].\nIt is *Noetherian* if it can be covered by finitely many such open sets.\n\n## Remarks\n\nA scheme is locally Noetherian if and only if $A$ is Noetherian for every open affine subscheme $\\operatorname{Spec} A$.\nSo $\\operatorname{Spec} A$ is Noetherian if and only if $A$ is a Noetherian ring.\nThe underlying space of a Noetherian scheme is a [[Noetherian topological space|2VQX2E8EAH]].\nExample: projective space $\\mathbb{P}^n_k$ over a field $k$ is Noetherian, being covered by $n+1$ copies of $\\operatorname{Spec} k[x_1, \\ldots, x_n]$, which are Noetherian by the Hilbert basis theorem.\nNon-example: $\\operatorname{Spec} k[x_1, x_2, \\ldots]$ is not Noetherian, since the ideal $(x_1, x_2, \\ldots)$ is not finitely generated.\nNon-example: a disjoint union of infinitely many copies of $\\operatorname{Spec} k$ is locally Noetherian but not Noetherian, since it is not quasi-compact.\n",
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