Local Homology and Local Cohomology
Let $(R, {\frak m})$ be a local ring, $I$ a proper ideal of $R$ and $M$ a finitely generated $R$-module of dimension $d$. We discuss the local homology modules of $H^d_I(M)$. When $M$ is Cohen-Macaulay, it is proved that $H^d_{{\frak m}}(M)$ is co-Cohen-Macaulay of N.dimension $d$ and $H^{\underline{x}}_d(H^d_{{\frak m}}(M))\cong\hat{M}$ where $\underline{x}=(x_1,...,x_d)$ is a system of parameters for $M$.
Some applications of a criterion for artinianness of a module
Abstract A criterion previously shown by the author for a module to be artinian is first applied to quickly deduce that certain local cohomology modules are artinian and thereafter to determine the first nonartinian local cohomology module using the concept of filter-depth. Next homomorphisms having finite index are considered. The logarithmic law is proved for such homomorphisms, and the index of an endomorphism on a noetherian (artinian) module is shown to be nonnegative (resp. nonpositive). Finally finiteness results for Koszul (co) homology modules are given, which are extensions of Matijevic's classical theorem on maximal ideal transforms.
Eine Dualisierung des Theorems von Matijevic
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