Sequentially Cohen-Macaulay matroidal ideals

Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and let $J$ be a matroidal ideal of degree $d$ in $R$. In this paper, we study the class of sequentially Cohen-Macaulay matroidal ideals. In particular, all sequentially Cohen-Macaulay matroidal ideals of degree $2$ are classified. Furthermore, we give a classification of sequentially Cohen-Macaulay matroidal ideals of degree $d\geq 3$ in some special cases.Comment: 12 pages, Comments are welcome

A note on graded generalized local cohomology modules

Let R = ⊕n∈N0Rn be a positively graded commutative Noetherian ring with irrelevant ideal R+ = ⊕n∈NRn and let a be a graded ideal contained in R+. Let M, N be two finitely generated graded R-modules. In this paper, we study the finiteness and vanishing of the n-th graded component Hia (M;N)n of the i-th generalized local cohomology module of M and N with respect to a. Also, we show that the least i such that R+ ⊊ √AnnR(Hia (M;N)) is equal to the least integer i for which Hia (M;N) is not finitely graded, where a graded module is finitely graded which is non-zero in only finitely many graded pieces.Keywords: Local cohomology modules, generalized local cohomology modules, graded module

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