Some Results for Formal Local Cohomology Modules

Let （R, m） be a commutative Noetherian local ring, I an ideal of R and M a finitely generated R-module. Let limnHm^i（M/I^nM）be the ith formal local cohomology module of M with respect to I.In this paper, we discuss some properties of formal local cohomology modules limnHm^i（M/I^nM）,which are analogous to the finiteness and Artinianness of local cohomology modules of a finitely generated module.

Cofiniteness of Local Cohomology Modules with Respect to a Pair of Ideals

Let R be a commutative Noetherian ring, I and J be two ideals of R, and M be an R-module. We study the cofiniteness and finiteness of the local cohomology module HiI,J（M） and give some conditions for the finiteness of HomR（R/I, HsI,J（M）） and Ext1R（R/I, HsI,J（M））. Also, we get some results on the attached primes of HdimMI,J （M）.

The Artinianness of Local Cohomology Modules

In this paper, let （R, m） be a Noetherian local ring, I lohtain in R an ideal, M and N be two finitely generated modules. Firstly, we study the properties of HI^t（M）, t = f-depth（I, M） and discuss the relationship between the Artinianness of HI^i（M, N） and the Artinianness of HI^i（N）. Then, we get that HI^d（M, N） is I-cofinite, if （R, m） is a d-dimensional Gorenstein local ring.

Local cohomology for Gorenstein homologically smooth DG algebras

In this paper,we introduce the theory of local cohomology and local duality to Notherian connected cochain DG algebras.We show that the notion of the local cohomology functor can be used to detect the Gorensteinness of a homologically smooth DG algebra.For any Gorenstein homologically smooth locally finite DG algebra A,we define a group homomorphism Hdet:Aut_(dg)(A)→k^(×),called the homological determinant.As applications,we present a sufficient condition for the invariant DG subalgebra A^(G)to be Gorenstein,where A is a homologically smooth DG algebra such that H(A)is a Noetherian AS-Gorenstein graded algebra and G is a finite subgroup of Aut_(dg)(A).Especially,we can apply this result to DG down-up algebras and non-trivial DG free algebras generated in two degree-one elements.

On the Finiteness Dimension of Local Cohomology Modules

Let R be a commutative Noetherian ring, α an ideal of R, and M a non-zero finitely generated R-module. Let t be a non-negative integer. In this paper, it is shown that dim Supp Hi a（M） ≤ 1 for all i 〈 t if and only if there exists an ideal b of R such that dimR/b ≤ 1 and Hia（M） ≌ Hi b（M） for all i 〈 t. Moreover, we prove that dimSuppHia（M） 〈≤dim M - i for all i.

Artinianness and Attached Primes of Formal Local Cohomology Modules

Let a be an ideal of a local ring （R, m） and A4 a finitely generated R-modu｜e. In this paper we study the Artinianness properties of formal local cohomology modules and we obtain the lower and upper bounds for Artinianness of formal local cohomology modules. Additionally, we determine the set AttR a dim M（M） and we show that the set of all non-isomorphic formal local cohomology modules a dim M（M） is finite.

Cofiniteness of Local Cohomology Modules

Let M be a non-zero finitely generated module over a commutative Noetherian local ring （R, m）. In this paper we consider when the local cohomology modules are finitely generated. It is shown that if t≥ 0 is an integer and p C Supp H^t_p （M）, then Hm^t＋dim R/p （M） is not p-cofinite. Then we obtain a partial answer to a question raised by Huneke. Namely, if R is a complete local ring, then H^n_m （M） is finitely generated if and only if 0 ≤ n ￠ W, where W ---- {t ＋ dimR/p丨p ∈ SuppH^t_p（M）／V（m）}. Also, we show that if J C I are 1-dimensional ideals of R, then H^t_I（M） is J-cominimax, and H^t_I（M） is finitely generated （resp., minimax） if and only if H}R, （Mp） is finitely generated for all p C Spec R （resp., p ∈ SpecR／MaxR）. Moreover, the concept of the J-cofiniteness dimension cJ（M） of M relative to I is introduced, and we explore an interrelation between c^I_m（M） and the filter depth of M in I. Finally, we show that if R is complete and dim M/IM ≠ 0, then c^I_m （R） ---- inf{depth Mp ＋ dim R/p 丨 P ∈ Supp M/IM／V（m）}.

Some Criteria for the Cohen-Macaulay Property and Local Cohomology

Let a be an ideal of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d. We characterize Cohen-Macaulay rings in term of a special homological dimension. Lastly, we prove that if R is a complete local ring, then the Matlis dual of top local cohomology module Ha^d（M） is a Cohen-Macaulay R-module provided that the R-module M satisfies some conditions.

Filter Depth and Cofiniteness of Local Cohomology Modules Defined by a Pair of Ideals

Let I, J be ideals of a commutative Noetherian local ring （R, m） and let M be a finite R-module. The f-depth of M with respect to I is the least integer r such that H_I（M） is not Artinian. In this paper we show that inf{f-depth（a, M） 丨a ∈ W（I, J）} is the least integer such that the local cohomology module with respect to a pair of ideals I, J is not Artinian. As a consequence, it follows that H_I,J（M） is （I, J）-cofinite for all i 〈 inf{f-depth（a, M） 丨a ∈ W（I, J）}. In addition, we show that for a Serre subcategory S, if H_I,J（M） belongs to S for all i 〉 n and if b is an ideal of R such that H^n_I,J（M/bM） belongs to S, then the module H^n_I,J（M）/bH^n_I,J（M） belongs to S.

Lower Bounds for Local Cohomology Modules with Respect to a Pair of Ideals

Let R be a Noetherian ring, I and J two ideals of R, M an R-module and t an integer. Let S be a Serre subcategory of the category of R-modules satisfying the co ondition CI, and N be a finitely generated R-module with SuppRN= V（a） for some a ∈ W（I, J）. It is shown that if ExtR^J（N,HI^i,J（M）） ∈ S for all i 〈 t and all j 〈 t - i, then Ha^i（M） ∈ S for all i 〈 t. Let S be the class of all R-modules N with divmR N ≤ k, where k is an integer. It is proved that if Ha^i（M） ∈ S for all i 〈 t and all a ∈ W（I, J）, then HI^i,j（M） ∈ S for all i 〈 t. It follows that inf{i ： HI^i,j（M） S} = inf（inf{i ： Ha^i（M） S）： a ∈W（I,J）}.

Tameness and Artinianness of Graded Generalized Local Cohomology Modules

Let R = ⊙n〉0 Rn be a standard graded ring, a ∩ ⊙n〉0 Rn an ideal of R, and M, N two finitely generated graded R-modules. This paper studies the homogeneous components of graded generalized local cohomology modules. We show that for any i 〉 0, the n-th graded component Hiα（M, N）n of the i-th generalized local cohomology module of M and N with respect to a vanishes for all n 〉〉 0. Some sufficient conditions are pro- posed to satisfy the equality sup{end（Hiα （M, N）） [ i _〉 0} = sup{end（HiR＋ （M, N）） ｜ i 〉 0}. Also, some sufficient conditions are proposed for the tameness of Hiα（M, N） such that i = fRα＋（M,N） or i = cdα（M,g）, where fRα＋（M,N） and cdα（M,g） denote the R＋- finiteness dimension and the cohomological dimension of M and N with respect to a, respectively. Finally, we consider the Artinian property of some submodules and quotient modules of Hjα（M, N）, where j is the first or last non-minimax level of Hiα（M, N）.

3redon-Illman cohomology with actions of ;otally disconnected groups

We study the Bredon-IUman cohomology with local coefficients for a G-space X in the case of G being a totally disconnected, locally compact group. We prove that any short exact sequence of equivariant local coefficients systems on X gives a long exact sequence of the associated Bredon-Illman cohomology groups with local coefficients.

TOWARD A THEORY OF WEAK I SEQUENCES

The author introduces a notion of weakIsequences and characterizes such sequences by means of homological methods.This notion extends the notion of weakMsequences and thus extends the notions of generalized Cohen Macaulay modules and Buchsbaum modules to more general cases.

A Note on Quadratically Parametrized Surfaces

Let f0, f1, f2, f3 be linearly independent homogeneous quadratic forms in the standard Z-graded ring R ：= K[s, t, u], and gcd（f0, f1, f2, f3） = 1. This defines a rational map Ф ： P2 → P3. The Rees algebra Rees（I） = R I I2 … of the ideal I = （f0, fl, f2, fs） is the graded R-algebra which can be described as the image of an R-algebra homomorphism h ： R[x, y, z, w] → Rees（I）. This paper discusses the free resolutions of I, and the structure of ker（h）.

On Standard Ideals

In the paper, we study a class of standard ideals which are more general than the m-primary standard ideals discussed in [2]. We will prove an important equality concerning I-weak sequences; thus a generalization of the equality of [2] is established.