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Matlis duality
In algebra, Matlis duality is a duality between Artinian and Noetherian modules over a complete Noetherian local ring. In the special case when the local ring has a field mapping to the residue field it is closely related to earlier work by Francis Sowerby Macaulay on polynomial rings and is sometimes called Macaulay duality, and the general case was introduced by .
==Statement==
Suppose that ''R'' is a Noetherian complete local ring with residue field ''k'', and choose ''E'' to be an injective hull of ''k'' (sometimes called a Matlis module). The dual ''D''''R''(''M'') of a module ''M'' is defined to be Hom''R''(''M'',''E''). Then Matlis duality states that the duality functor ''D''''R'' gives an anti-equivalence between the categories of Artinian and Noetherian ''R''-modules. In particular the duality functor gives an anti-equivalence from the category of finite-length modules to itself.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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