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 $M$.
Co-Cohen-Macaulay Modules and Local Cohomology
Let be a commutative Noetherian local ring and let be a finitely generated -module of dimension . Then the following statements hold: (a) if width for all with , then is co-Cohen-Macaulay of Noetherian dimension ; (b) if is an unmixed -module and depth , then is co-Cohen-Macaulay of Noetherian dimension if and only if is either zero or co-Cohen-Macaulay of Noetherian dimension . As consequence, if is co-Cohen-Macaulay of Noetherian dimension for all with , then is co-Cohen-Macaulay of Noetherian dimension .
Adic-completion and some dual homological results
Let a be an ideal of a commutative ring A. There is a kind of duality between the left derived functors Uia of the a-adic completion functor, called local homology functors, and the local cohomology functors Hai.
Some dual results are obtained for these Uia, and also inequalities involving both local homology and local cohomology when the ring A is noetherian or more generally when the Ua and Ha-global dimensions of A are finite.