Local Duality for Modules over Noetherian Commutative Rings

Abstract

Some applications of the general theorem on the existence of local duality for modules over Noetherian commutative rings are given. Let $$\Lambda$$ be a Noetherian commutative ring, let $$\mathcal{M} = \left\{ \mathfrak{M} \right\}$$ be a set of maximal ideals in $$\Lambda$$ , and let $$\widehat\Lambda = \underleftarrow {\lim }\,\Lambda _\mathfrak{M} ,\Gamma \left( \Lambda \right) = \prod\limits_{\mathfrak{M} \in \mathcal{M}} {\widehat\Lambda _\mathfrak{M} }$$ . Then the category of Artinian modules is dual to the category of Noetherian modules. Several structural results are proved, including the theorem on the structure of Artinian modules over principal ideal domains. For rings of special kinds, double centralizer theorems are proved. Bibliography: 5 titles.