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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク equivariant sheaf ： ウィキペディア英語版 equivariant sheaf In mathematics, given the action $\backslash sigma:\; G\; \backslash times\_S\; X\; \backslash to\; X$ of a group scheme ''G'' on a scheme (or stack) ''X'' over a base scheme ''S'', an equivariant sheaf ''F'' on ''X'' is a sheaf of $\backslash mathcal\_X$-modules together with the isomorphism of $\backslash mathcal\_$-modules :$\backslash phi:\; \backslash sigma^$ * F \simeq p_2^ *F   that satisfies the cocycle condition: writing ''m'' for multiplication, :$p\_^$ * \phi \circ (1_G \times \sigma)^ * \phi = (m \times 1_X)^ * \phi. On the stalk level, the cocycle condition says that the isomorphism $F\_\; \backslash simeq\; F\_x$ is the same as the composition $F\_\; \backslash simeq\; F\_\; \backslash simeq\; F\_x$; i.e., the associativity of the group action. The unitarity of a group action, on the other hand, is a consequence: applying $(e\; \backslash times\; e\; \backslash times\; 1)^$ *, e: S \to G to both sides gives $(e\; \backslash times\; 1)^$ * \circ (e \times 1)^ * \phi = (e \times 1)^ * \phi and so $(e\; \backslash times\; 1)^$ * \phi is the identity. Note that $\backslash phi$ is an additional data; it is "a lift" of the action of ''G'' on ''X'' to the sheaf ''F''. A structure of an equivariant sheaf on a sheaf (namely $\backslash phi$) is also called a linearlization. In practice, one typically imposes further conditions; e.g., ''F'' is quasi-coherent, ''G'' is smooth and affine. If the action of ''G'' is free, then the notion of an equivariant sheaf simplifies to a sheaf on the quotient ''X''/''G'', because of the descent along torsors. By Yoneda's lemma, to give the structure of an equivariant sheaf to an $\backslash mathcal\_X$-module ''F'' is the same as to give group homomorphisms for rings ''R'' over $S$, :$G(R)\; \backslash to\; \backslash operatorname(X\; \backslash times\_S\; \backslash operatornameR,\; F\; \backslash otimes\_S\; R)$. Remark: There is also a definition of equivariant sheaves in terms of simplicial sheaves. One example of an equivariant sheaf is a linearlized line bundle in geometric invariant theory. Another example is the sheaf of equivariant differential forms. == Equivariant vector bundle == A definition is simpler for a vector bundle (i.e., a variety corresponding to a locally free sheaf of constant rank). We say a vector bundle ''E'' on an algebraic variety ''X'' acted by an algebraic group ''G'' is ''equivariant'' if ''G'' acts fiberwise: i.e., $g:\; E\_x\; \backslash to\; E\_$ is a "linear" isomorphism of vector spaces.〔If ''E'' is viewed as a sheaf, then ''g'' needs to replaced by $g^$.〕 In other words, an equivariant vector bundle is a pair consisting of a vector bundle and the lifting of the action $G\; \backslash times\; X\; \backslash to\; X$ to that of $G\; \backslash times\; E\; \backslash to\; E$ so that the projection $E\; \backslash to\; X$ is equivariant. (Locally free sheaves and vector bundles correspond contravariantly. Thus, if ''V'' is a vector bundle corresponding to ''F'', then $\backslash phi$ induces isomorphisms between fibers $V\_x\; \backslash overset\backslash to\; V\_$, which are linear maps.) Just like in the non-equivariant setting, one can define an equivariant characteristic class of an equivariant vector bundle. 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「equivariant sheaf」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.