. . . "En g\u00E9om\u00E9trie alg\u00E9brique, un morphisme de sch\u00E9mas peut \u00EAtre vu comme une famille de sch\u00E9mas param\u00E9tr\u00E9e par les points de Y. La notion de platitude de f est une sorte de continuit\u00E9 de cette famille."@fr . . "\u5E73\u5766\u5C04"@ja . . "\u5E73\u5766\u5C04\uFF08\u3078\u3044\u305F\u3093\u3057\u3083\u3001\u82F1: flat morphism\uFF09\u3068\u306F\u3001\u6570\u5B66\u306E\u4EE3\u6570\u5E7E\u4F55\u5B66\u306B\u304A\u3051\u308B\u30B9\u30AD\u30FC\u30E0\u8AD6\u306E\u7528\u8A9E\u3067\u3001\u30B9\u30AD\u30FC\u30E0 X \u304B\u3089\u30B9\u30AD\u30FC\u30E0 Y \u3078\u306E\u5C04f\u3067\u3042\u3063\u3066\u830E\u306B\u8A98\u5C0E\u3055\u308C\u308B\u5199\u50CF\u304C\u3059\u3079\u3066\u74B0\u306E\u5E73\u5766\u5199\u50CF\u306B\u306A\u308B\u3082\u306E\u306E\u3053\u3068\u3092\u3044\u3046\u3002\u3064\u307E\u308A\u3001X \u306E\u3059\u3079\u3066\u306E\u70B9 P \u306B\u5BFE\u3057\u3066 \u304C\u5E73\u5766\u5199\u50CF\u306B\u306A\u308B\u3082\u306E\u306E\u3053\u3068\u3092\u3044\u3046\u3002\u74B0\u306E\u5199\u50CF\u304C\u5E73\u5766\u3068\u306F\u3001\u6E96\u540C\u578B\u3067\u3042\u3063\u3066\u3053\u308C\u306B\u3088\u308AB\u304C\u5E73\u5766 A \u52A0\u7FA4\u306B\u306A\u308B\u3053\u3068\u3067\u3042\u308B\u3002\u30B9\u30AD\u30FC\u30E0\u306E\u5C04\u304C\u5168\u5C04\u304B\u3064\u5E73\u5766\u3067\u3042\u308B\u3068\u304D\u3001\u5FE0\u5B9F\u5E73\u5766\u3068\u3044\u3046\u3002 \u5E73\u5766\u5C04\u306E\u611F\u899A\u7684\u306A\u7406\u89E3\u306E\u3046\u3048\u3067\u306F\u6B21\u306E2\u3064\u304C\u57FA\u672C\u7684\u3067\u3042\u308B\u3002 \n* \u5E73\u5766\u6027\u306F\u3067\u3042\u308B\uFF08\u3053\u306E\u3053\u3068\u3092\u4EE5\u4E0B\u3067\u306F\u3068\u547C\u3076\uFF09\u3002\u3064\u307E\u308A\u3001\uFF08\u3042\u308B\u6709\u9650\u6027\u306E\u6761\u4EF6\u306E\u4E0B\u3067\uFF09\u30B9\u30AD\u30FC\u30E0\u306E\u5C04\u306F\u307B\u3068\u3093\u3069\u306E\u70B9\u3067\u5E73\u5766\u3067\u3042\u308A\u3001\u5E73\u5766\u6027\u304C\u5D29\u308C\u308B\u306E\u306F\u4F8B\u5916\u7684\u306A\u90E8\u5206\u96C6\u5408\u306B\u304A\u3044\u3066\u3067\u3042\u308B\u3002\u3053\u306E\u3053\u3068\u306F\u53EF\u63DB\u74B0\u8AD6\u306B\u304A\u3051\u308B\u4E00\u822C\u81EA\u7531\u6027\u306E\u5E30\u7D50\u3067\u3042\u308B\u3002 \n* \u5E73\u5766\u5C04\u3067\u306F\u30D5\u30A1\u30A4\u30D0\u30FC\u306E\u7B49\u6B21\u5143\u6027\u304C\u6210\u308A\u7ACB\u3064\u3002\u307E\u305F\u3001\u3042\u308B\u4EEE\u5B9A\u306E\u3082\u3068\u3067\u306F\u30D5\u30A1\u30A4\u30D0\u30FC\u304C\u7B49\u6B21\u5143\u3067\u3042\u308C\u3070\u5E73\u5766\u3067\u3042\u308B\uFF08\uFF09\u3002\u3053\u306E\u3053\u3068\u304B\u3089\u3001\u5E73\u5766\u6027\u3068\u306F\u3059\u306A\u308F\u3061\u30D5\u30A1\u30A4\u30D0\u30FC\u304C\u7B49\u6B21\u5143\u3067\u3042\u308B\u3053\u3068\u3068\u601D\u3048\u308B\u3002\u3053\u306E\u70B9\u306B\u7279\u306B\u7740\u76EE\u3057\u3001\u30D5\u30A1\u30A4\u30D0\u30FC\u306B\u7B49\u6B21\u5143\u6027\u306E\u6761\u4EF6\u3092\u8AB2\u3057\u305F\u3044\u3068\u304D\u306B\u5E73\u5766\u6027\u3092\u4EEE\u5B9A\u3059\u308B\u3053\u3068\u304C\u3042\u308B\u3002\u307E\u305F\u3001\u5E73\u5766\u5C04\u306F\u7B49\u6B21\u5143\u306E\u30D5\u30A1\u30A4\u30D0\u30FC\u306E\u65CF\u3067\u3042\u308B\u3053\u3068\u3092\u5F37\u8ABF\u3057\u305F\u3044\u3068\u304D\u3001\u5E73\u5766\u5C04\u3092\u5E73\u5766\u65CF\u3068\u3044\u3046\u3053\u3068\u3082\u591A\u3044\u3002\u4F8B\u3048\u3070\u3001\u53CC\u6709\u7406\u5E7E\u4F55\u5B66\u3067\u306E\u4EE3\u6570\u66F2\u9762\u306E\u30D6\u30ED\u30FC\u30C0\u30A6\u30F3\u3068\u3044\u3046\u64CD\u4F5C\u3067\u306F\u3001\u3042\u308B\u7279\u5B9A\u306E1\u70B9\u3067\u306E\u306E\u6B21\u5143\u306F1\u3067\u3042\u308B\u304C\u3001\u4ED6\u306E\u70B9\u3067\u306E\u6B21\u5143\u306F\u3059\u3079\u30660\u306A\u306E\u3067\u3001\u5E73\u5766\u3067\u306F\u306A\u3044\u3002 \u5E73\u5766\u5C04\u306F\u8272\u3005\u306A\u7A2E\u985E\u306E\u3084\u306E\u5B9A\u7FA9\u306B\u4F7F\u308F\u308C\u308B\u3002\u3053\u308C\u3089\u306F\u6DF1\u3044\u7406\u8AD6\u3067\u3001\u6271\u3044\u3084\u3059\u3044\u3082\u306E\u3067\u306F\u306A\u3044\u3002\u5E73\u5766\u5C04\u306F\u30A8\u30BF\u30FC\u30EB\u5C04\u306E\u5B9A\u7FA9\u3001\u3072\u3044\u3066\u306F\u30A8\u30BF\u30FC\u30EB\u30FB\u30B3\u30DB\u30E2\u30ED\u30B8\u30FC\u306E\u5B9A\u7FA9\u306B\u3082\u4F7F\u308F\u308C\u308B\u3002\u30A8\u30BF\u30FC\u30EB\u5C04\u3068\u306F\u3001\u5E73\u5766\u304B\u3064\u6709\u9650\u578B\u304B\u3064\u4E0D\u5206\u5C90\u306A\u5C04\u306E\u3053\u3068\u3067\u3042\u3063\u305F\u3002"@ja . . "1091504021"^^ . . . . . . . . . . "542520"^^ . . . . . . . "In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e., is a flat map for all P in X. A map of rings is called flat if it is a homomorphism that makes B a flat A-module. A morphism of schemes is called faithfully flat if it is both surjective and flat. Two basic intuitions regarding flat morphisms are: \n* flatness is a generic property; and \n* the failure of flatness occurs on the jumping set of the morphism."@en . . "\uD3C9\uD0C4 \uC0AC\uC0C1"@ko . . . . . . . . "Flat morphism"@en . "\u5E73\u5766\u5C04\uFF08\u3078\u3044\u305F\u3093\u3057\u3083\u3001\u82F1: flat morphism\uFF09\u3068\u306F\u3001\u6570\u5B66\u306E\u4EE3\u6570\u5E7E\u4F55\u5B66\u306B\u304A\u3051\u308B\u30B9\u30AD\u30FC\u30E0\u8AD6\u306E\u7528\u8A9E\u3067\u3001\u30B9\u30AD\u30FC\u30E0 X \u304B\u3089\u30B9\u30AD\u30FC\u30E0 Y \u3078\u306E\u5C04f\u3067\u3042\u3063\u3066\u830E\u306B\u8A98\u5C0E\u3055\u308C\u308B\u5199\u50CF\u304C\u3059\u3079\u3066\u74B0\u306E\u5E73\u5766\u5199\u50CF\u306B\u306A\u308B\u3082\u306E\u306E\u3053\u3068\u3092\u3044\u3046\u3002\u3064\u307E\u308A\u3001X \u306E\u3059\u3079\u3066\u306E\u70B9 P \u306B\u5BFE\u3057\u3066 \u304C\u5E73\u5766\u5199\u50CF\u306B\u306A\u308B\u3082\u306E\u306E\u3053\u3068\u3092\u3044\u3046\u3002\u74B0\u306E\u5199\u50CF\u304C\u5E73\u5766\u3068\u306F\u3001\u6E96\u540C\u578B\u3067\u3042\u3063\u3066\u3053\u308C\u306B\u3088\u308AB\u304C\u5E73\u5766 A \u52A0\u7FA4\u306B\u306A\u308B\u3053\u3068\u3067\u3042\u308B\u3002\u30B9\u30AD\u30FC\u30E0\u306E\u5C04\u304C\u5168\u5C04\u304B\u3064\u5E73\u5766\u3067\u3042\u308B\u3068\u304D\u3001\u5FE0\u5B9F\u5E73\u5766\u3068\u3044\u3046\u3002 \u5E73\u5766\u5C04\u306E\u611F\u899A\u7684\u306A\u7406\u89E3\u306E\u3046\u3048\u3067\u306F\u6B21\u306E2\u3064\u304C\u57FA\u672C\u7684\u3067\u3042\u308B\u3002 \u5E73\u5766\u5C04\u306F\u8272\u3005\u306A\u7A2E\u985E\u306E\u3084\u306E\u5B9A\u7FA9\u306B\u4F7F\u308F\u308C\u308B\u3002\u3053\u308C\u3089\u306F\u6DF1\u3044\u7406\u8AD6\u3067\u3001\u6271\u3044\u3084\u3059\u3044\u3082\u306E\u3067\u306F\u306A\u3044\u3002\u5E73\u5766\u5C04\u306F\u30A8\u30BF\u30FC\u30EB\u5C04\u306E\u5B9A\u7FA9\u3001\u3072\u3044\u3066\u306F\u30A8\u30BF\u30FC\u30EB\u30FB\u30B3\u30DB\u30E2\u30ED\u30B8\u30FC\u306E\u5B9A\u7FA9\u306B\u3082\u4F7F\u308F\u308C\u308B\u3002\u30A8\u30BF\u30FC\u30EB\u5C04\u3068\u306F\u3001\u5E73\u5766\u304B\u3064\u6709\u9650\u578B\u304B\u3064\u4E0D\u5206\u5C90\u306A\u5C04\u306E\u3053\u3068\u3067\u3042\u3063\u305F\u3002"@ja . . . "Morphisme plat"@fr . . "20645"^^ . . . . . . . . . . . . . . . . . "In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e., is a flat map for all P in X. A map of rings is called flat if it is a homomorphism that makes B a flat A-module. A morphism of schemes is called faithfully flat if it is both surjective and flat. Two basic intuitions regarding flat morphisms are: \n* flatness is a generic property; and \n* the failure of flatness occurs on the jumping set of the morphism. The first of these comes from commutative algebra: subject to some on f, it can be shown that there is a non-empty open subscheme of Y, such that f restricted to Y\u2032 is a flat morphism (generic flatness). Here 'restriction' is interpreted by means of the fiber product of schemes, applied to f and the inclusion map of into Y. For the second, the idea is that morphisms in algebraic geometry can exhibit discontinuities of a kind that are detected by flatness. For instance, the operation of blowing down in the birational geometry of an algebraic surface, can give a single fiber that is of dimension 1 when all the others have dimension 0. It turns out (retrospectively) that flatness in morphisms is directly related to controlling this sort of semicontinuity, or one-sided jumping. Flat morphisms are used to define (more than one version of) the flat topos, and flat cohomology of sheaves from it. This is a deep-lying theory, and has not been found easy to handle. The concept of \u00E9tale morphism (and so \u00E9tale cohomology) depends on the flat morphism concept: an \u00E9tale morphism being flat, of finite type, and unramified."@en . . . . . "En g\u00E9om\u00E9trie alg\u00E9brique, un morphisme de sch\u00E9mas peut \u00EAtre vu comme une famille de sch\u00E9mas param\u00E9tr\u00E9e par les points de Y. La notion de platitude de f est une sorte de continuit\u00E9 de cette famille."@fr . .