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 $M$.
Tight Closure of Certain Submodules of the Top Local Cohomology
. Let ( R, m ) be a d -dimensional Cohen-Macaulay local ring of prime characteristic. Then the top local cohomology module H dm ( R ) carries important information of R . So we will study the tight closure of submodules of H dm ( R ). It is shown that R is weakly F -regular if every ﬁnitely generated submodule of H dm ( R ) is tightly closed in H dm ( R ). The colon capturing properties of tight closure of certain submodules of H dm ( R ) are also obtained.
The Globalization problem for inner automorphisms and Skolem-Noether theorems
We investigate the circumstances under which an inner automorphism of a ring with local units can be built from “local” information. Specifically, we consider three natural “inner” type properties for an automorphism of a ring with local units. We show that every inner automorphism is locally inner but the converse is false, even if the automorphism is “piecewise” inner. On the other hand, we construct a large class of rings for which every locally inner automorphism is actually inner. Finally we obtain some Skolem-Noether type Theorems for infinite matrix and triangular matrix rings.