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In abstract algebra, a dualizing module, also called a canonical module, is a module over a commutative ring that is analogous to the canonical bundle of a smooth variety. It is used in Grothendieck local duality. ==Definition== A dualizing module for a Noetherian ring ''R'' is a finitely generated module ''M'' such that for any maximal ideal ''m'', the ''R''/''m'' vector space vanishes if ''n'' ≠ height(''m'') and is 1-dimensional if ''n'' = height(''m''). A dualizing module need not be unique because the tensor product of any dualizing module with a rank 1 projective module is also a dualizing module. However this is the only way in which the dualizing module fails to be unique: given any two dualizing modules, one is isomorphic to the tensor product of the other with a rank 1 projective module. In particular if the ring is local the dualizing module is unique up to isomorphism. A Noetherian ring does not necessarily have a dualizing module. Any ring with a dualizing module must be Cohen–Macaulay. Conversely if a Cohen–Macaulay ring is a quotient of a Gorenstein ring then it has a dualizing module. In particular any complete local Cohen–Macaulay ring has a dualizing module. For rings without a dualizing module it is sometimes possible to use the dualizing complex as a substitute. 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「Dualizing module」の詳細全文を読む スポンサード リンク
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