Eine Charakterisierung der Matlis-reflexiven Moduln

TL;DR

This paper characterizes Matlis-reflexive modules over noetherian local rings using Bass numbers, providing criteria for reflexivity based on module homomorphisms and duality properties.

Contribution

It establishes a new criterion for reflexivity of modules via Bass numbers and explores implications for modules with monomorphisms or epimorphisms involving the injective hull.

Findings

Reflexivity characterized by Bass numbers equality for all primes.

Modules with monomorphisms or epimorphisms involving the injective hull are reflexive.

Provides a criterion for reflexivity based on module homomorphisms.

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.