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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 Stable module category ： ウィキペディア英語版 Stable module category In representation theory, the stable module category is a category in which projectives are "factored out." == Definition == Let ''R'' be a ring. For two modules ''M'' and ''N'', define $\backslash underline\}(M,N)$. Given a module ''M'', let ''P'' be a projective module with a surjection $p\; \backslash colon\; P\; \backslash to\; M$. Then set $\backslash Omega(M)$ to be the kernel of ''p''. Suppose we are given a morphism $f\; \backslash colon\; M\; \backslash to\; N$ and a surjection $q\; \backslash colon\; Q\; \backslash to\; N$ where ''Q'' is projective. Then one can lift ''f'' to a map $P\; \backslash to\; Q$ which maps $\backslash Omega(M)$ into $\backslash Omega(N)$. This gives a well-defined functor $\backslash Omega$ from the stable module category to itself. For certain rings, such as Frobenius algebras, $\backslash Omega$ is an equivalence of categories. In this case, the inverse $\backslash Omega^$ can be defined as follows. Given ''M'', find an injective module ''I'' with an inclusion $i\; \backslash colon\; M\; \backslash to\; I$. Then $\backslash Omega^(M)$ is defined to be the cokernel of ''i''. A case of particular interest is when the ring ''R'' is a group algebra. The functor Ω−1 can even be defined on the module category of a general ring (without factoring out projectives), as the cokernel of the injective envelope. It need not be true in this case that the functor Ω−1 is actually an inverse to Ω. One important property of the stable module category is it allows defining the Ω functor for general rings. When ''R'' is perfect (or ''M'' is finitely generated and ''R'' is semiperfect), then Ω(''M'') can be defined as the kernel of the projective cover, giving a functor on the module category. However, in general projective covers need not exist, and so passing to the stable module category is necessary. 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「Stable module category」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.