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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0310REJJY8]].\nThen the [[spectrum|FB45BHHA2J]] $\\operatorname{Spec} R$ with the [[Zariski topology|8SQK7RTB63]] is quasi-compact: every cover of $\\operatorname{Spec} R$ by open sets has a finite subcover.\n\n## Remarks\n\nIt is enough to treat covers by [[basic open sets|N6JX9BCKQP]] $D(f_i) = \\set{\\mathfrak p : f_i \\notin \\mathfrak p}$, $i \\in I$.\nThese cover $\\operatorname{Spec} R$ exactly when no prime ideal contains all $f_i$.\nSince every proper [[ideal|096Z9MP2ME]] lies in a [[maximal ideal|0PV74TR5FP]], this means that the $f_i$ generate the unit ideal.\nThen $1 = a_1 f_{i_1} + \\cdots + a_n f_{i_n}$ for finitely many indices, and $D(f_{i_1}), \\ldots, D(f_{i_n})$ already cover.\n\nExample: $D(2)$ and $D(3)$ cover $\\operatorname{Spec} \\Z$, because $1 = 3 - 2$.\n\nThe word \"quasi-compact\" is used because the space is usually not [[Hausdorff|0S1FJ8VEX9]], and some authors reserve \"[[compact|0TVJZYB6SH]]\" for Hausdorff spaces.\nThe same argument shows that each basic open set $D(f)$ is quasi-compact, since $D(f) \\cong \\operatorname{Spec} R_f$.\n\n## Explanations\n\n### L1 human\n\nHowever you cover the spectrum of a ring with open sets, a finite number of them already do the job.\nThe reason is algebraic: covering the space means producing $1$ from the given elements, and any such expression uses only finitely many of them.\n\n### L2 human\n\nQuasi-compactness of $\\operatorname{Spec} R$ is the algebraic shadow of the fact that $1$ is a finite sum.\nIt is used to pass from local to global data in finitely many steps.\nFor example, it lets one glue finitely many local constructions, or bound a quantity over a finite cover.\nIn scheme theory, it says that affine schemes are quasi-compact.\nIt is one of the conditions in [[Hochster's theorem on spectral spaces|C82AXZK2BS]].\n",
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