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In mathematics, Serre and localizing subcategories form important classes of subcategories of an abelian category. Localizing subcategories are certain Serre subcategories. They are strongly linked to the notion of a quotient category. == Serre subcategories == Let be an abelian category. A non-empty full subcategory is called a ''Serre subcategory'' (or also a ''dense subcategory''), if for every short exact sequence in the object is in if and only if the objects and belong to . In words: is closed under subobjects, quotient objects and extensions. The importance of this notion stems from the fact that kernels of exact functors between abelian categories have this property, and that one can build (for locally small ) the quotient category (in the sense of Gabriel, Grothendieck, Serre) , which has the same objects as , is abelian, and comes with an exact functor (called the quotient functor) whose kernel is . 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「Localizing subcategory」の詳細全文を読む スポンサード リンク
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