139 research outputs found
On the Structure of Sequentially Generalized Cohen-Macaulay Modules
A finitely generated module over a local ring is called a sequentially
generalized Cohen-Macaulay module if there is a filtration of submodules of
: such that and each is generalized Cohen-Macaulay. The aim
of this paper is to study the structure of this class of modules. Many basic
properties of these modules are presented and various characterizations of
sequentially generalized Cohen-Macaulay property by using local cohomology
modules, theory of multiplicity and in terms of systems of parameters are
given. We also show that the notion of dd-sequences defined in \cite{cc} is an
important tool for studying this class of modules.Comment: 28 page
On the finiteness and stability of certain sets of associated primes ideals of local cohomology modules
Let be a Noetherian local ring, an ideal of and a
finitely generated -module. Let be an integer and
r=\depth_k(I,N) the length of a maximal -sequence in dimension in
defined by M. Brodmann and L. T. Nhan ({Comm. Algebra, 36 (2008), 1527-1536).
For a subset S\subseteq \Spec R we set S_{{\ge}k}={\p\in
S\mid\dim(R/\p){\ge}k}. We first prove in this paper that
\Ass_R(H^j_I(N))_{\ge k} is a finite set for all }. Let
\fN=\oplus_{n\ge 0}N_n be a finitely generated graded \fR-module, where
\fR is a finitely generated standard graded algebra over . Let be
the eventual value of \depth_k(I,N_n). Then our second result says that for
all the sets \bigcup_{j{\le}l}\Ass_R(H^j_I(N_n))_{{\ge}k} are
stable for large .Comment: To appear in Communication in Algebr
A local homology theory for linearly compact modules
We introduce a local homology theory for linearly compact modules which is in
some sense dual to the local cohomology theory of A. Grothendieck. Some basic
properties such as the noetherianness, the vanishing and non-vanishing of local
homology modules of linearly compact modules are proved. A duality theory
between local homology and local cohomology modules of linearly compact modules
is developed by using Matlis duality and Macdonald duality. As consequences of
the duality theorem we obtain some generalizations of well-known results in the
theory of local cohomology for semi-discrete linearly compact modules.Comment: 24 page
A Splitting Theorem for Local Cohomology and its Applications
Let be a commutative Noetherian ring and a finitely generated
-module. We show in this paper that, for an integer , if the local
cohomology module with respect to an ideal is
finitely generated for all , then H^{i}_\mathfrak{a}(M/xM)\cong
H^{i}_\mathfrak{a}(M)\oplus H^{i+1}_\mathfrak{a}(M)\frak ax\frak ai<t-1$. As consequences we obtain generalizations, by very short proofs, of
the main results of M. Brodmann and A.L. Faghani (A finiteness result for
associated primes of local cohomology modules, Proc. Amer. Math. Soc.,
128(2000), 2851-2853) and of H.L. Truong and the first author (Asymptotic
behavior of parameter ideals in generalized Cohen-Macaulay module, J. Algebra,
320(2008),158-168).Comment: to appear in J. Algebr
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