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In mathematics, the Gabriel–Popescu theorem is an embedding theorem for certain abelian categories, introduced by . It characterizes certain abelian categories (the Grothendieck categories) as quotients of module categories. There are several generalizations and variations of the Gabriel–Popescu theorem, given by (for an AB5 category with a set of generators), , (for triangulated categories). ==Theorem== Let ''A'' be a Grothendieck category (an AB5 category with a generator), ''U'' a generator of ''A'' and ''R'' be the ring of endomorphisms of ''U''; also, let ''S'' be the functor from ''A'' to Mod-''R'' (the category of right ''R''-modules) defined by ''S''(''X'') = Hom(''U'',''X''). Then the Gabriel–Popescu theorem states that ''S'' is full and faithful and has an exact left adjoint. This implies that ''A'' is equivalent to the Serre quotient category of Mod-''R'' by a certain localizing subcategory ''C''. (A localizing subcategory of Mod-''R'' is a full subcategory ''C'' of Mod-''R'', closed under arbitrary direct sums, such that for any short exact sequence of modules , we have ''M2'' in ''C'' if and only if ''M1'' and ''M3'' are in ''C''. The Serre quotient of Mod-''R'' by any localizing subcategory is a Grothendieck category.) We may take ''C'' to be the kernel of the left adjoint of the functor ''S''. 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「Gabriel–Popescu theorem」の詳細全文を読む スポンサード リンク
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