翻訳と辞書 Words near each other ・ Cohoes ・ Cohoes City Hall ・ Cohoes Commons ・ Cohoes Falls ・ Cohoes High School ・ Cohoes Music Hall ・ Cohoes, New York ・ Cohoke, Virginia ・ Cohomological descent ・ Cohomological dimension ・ Cohomological invariant ・ Cohomology ・ Cohomology of algebras ・ Cohomology operation ・ Cohomology ring ・ Cohomology with compact support ・ Cohomotopy group ・ Cohong ・ Cohoni ・ Cohonina ・ Cohons ・ Cohoquinoque Creek ・ Cohors amicorum ・ Cohors I Aelia Dacorum ・ Cohors I Aelia Gaesatorum milliaria sagitt ・ Cohors I Alpinorum equitata ・ Cohors I Alpinorum peditata ・ Cohors I Antiochensium equitata ・ Cohors I Aquitanorum ・ Cohors I Aquitanorum veterana Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cohomology with compact support ： ウィキペディア英語版 Cohomology with compact support In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support. ==de Rham cohomology with compact support for smooth manifolds== Given a manifold ''X'', let $\backslash Omega^k\_(X)$ be the real vector space of ''k''-forms on ''X'' with compact support, and ''d'' be the standard exterior derivative. Then the de Rham cohomology groups with compact support $H^q\_(X)$ are the homology of the chain complex $(\backslash Omega^\backslash bullet\_(X),d)$: :$0\; \backslash to\; \backslash Omega^0\_(X)\; \backslash to\; \backslash Omega^1\_(X)\; \backslash to\; \backslash Omega^2\_(X)\; \backslash to\; \backslash cdots$ ''i.e.'', $H^q\_(X)$ is the vector space of closed ''q''-forms modulo that of exact ''q''-forms. Despite their definition as the homology of an ascending complex, the de Rham groups with compact support demonstrate covariant behavior; for example, given the inclusion mapping ''j'' for an open set ''U'' of ''X'', extension of forms on ''U'' to ''X'' (by defining them to be 0 on ''X''–''U'') is a map $j\_$ *: \Omega^\bullet_(U) \to \Omega^\bullet_(X) inducing a map :$j\_$ *: H^q_(U) \to H^q_(X). They also demonstrate contravariant behavior with respect to proper maps - that is, maps such that the inverse image of every compact set is compact. Let ''f'': ''Y'' → ''X'' be such a map; then the pullback :$f^$ *: \Omega^q_(X) \to \Omega^q_(Y) \sum_I g_I \, dx_ \wedge \ldots \wedge dx_ \mapsto \sum_I(g_I \circ f) \, d(x_ \circ f) \wedge \ldots \wedge d(x_ \circ f) induces a map :$H^q\_(X)\; \backslash to\; H^q\_(Y)$. If ''Z'' is a submanifold of ''X'' and ''U'' = ''X''–''Z'' is the complementary open set, there is a long exact sequence :$\backslash cdots\; \backslash to\; H^q\_(U)\; \backslash overset\; H^q\_(X)\; \backslash overset\; H^q\_(Z)\; \backslash overset\; H^\_(U)\; \backslash to\; \backslash cdots$ called the long exact sequence of cohomology with compact support. It has numerous applications, such as the Jordan curve theorem, which is obtained for ''X'' = R² and ''Z'' a simple closed curve in ''X''. De Rham cohomology with compact support satisfies a covariant Mayer–Vietoris sequence: if ''U'' and ''V'' are open sets covering ''X'', then :$\backslash cdots\; \backslash to\; H^q\_(U\; \backslash cap\; V)\; \backslash to\; H^q\_(U)\backslash oplus\; H^q\_(V)\; \backslash to\; H^q\_(X)\; \backslash overset\; H^\_(U\backslash cap\; V)\; \backslash to\; \backslash cdots$ where all maps are induced by extension by zero is also exact. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cohomology with compact support」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.