Eine Charakterisierung der Matlis-reflexiven Moduln

Abstract

Let $(R,\my)$ be a noetherian local ring, $E$ the injective hull of $k=R/\my$ and $M^\circ=$ Hom$_R(M,E)$ the Matlis dual of the $R$-module $M$. If the canonical monomorphism $\varphi: M \to \moo$ is surjective, $M$ is known to be called (Matlis-)reflexive. With the help of the Bass numbers $\mu(\py,M)=\dim_{\kappa(\py)}($Hom$_R(R/\py,M)_\py)$ of $M$ with respect to $\py$ we show: $M$ is reflexive if and only if $\mu(\py,M)=\mu(\py,\moo)$ for all $\py \in $ Spec$(R)$. From this it follows for every $R$-module $M$: If there exists a monomorphism $\moo \hookrightarrow M$ or an epimorphism $M \twoheadrightarrow \moo$, then $M$ is already reflexive.