مدول های کوهمولوژی موضعی

 Let a  be an ideal of Noetherian ring R  and M,N  be finitely generated R -modules. Recall that, for each i≥0 , i-th generalized local cohomology module M, N with respect to a  is defined byHaiM,N≔limn Hai(M/anM,N).Also, recall that  cda,M,N,  the cohomological dimension of R-modules M and N with respect to an ideal a  of a commutative Noetherian ring R  issupi∈N0: Hai M,N≠0.An important problem concerning local cohomology is determining the set of attached prime ideals of the top local cohomology modules. This problem has been studied by several authors. In this paper, we study attached prime ideals of top local cohomology modules.

 In this paper, we first obtain some subsets of the set of attached prime ideals of top local cohomology module. By using these, we obtain a result about finiteness of  top local cohomology modules.

  Let R  be a Noetherian ring and a  be an ideal of  R . Let M  and N  be non-zero finitely generated R -modules.  Assume that  pdM=d<∞ , cda,N=c<∞ . We will prove that i) p∈SuppRN:  cda,R/p=c, dimRp=c⊆AttR(HacN),ii)  AttR(Had+cM,N)⊆p∈SuppRN:  cda,M,R/p=d+c,iii) p∈SuppRN :cda,M,Rp=d+c,dimRp=c ⊆AttR(Had+cM,N).

 Let M  and N  be non-zero finitely generated R -modules.  Assume that  pdM=d<∞ , cda,N=c<∞.  The following conclusions were drawn from this research.  If  (R,m)   is a Noetherian local ring such that   Had+cM,N≠0  then Had+cM,N  is not of finite length. If R is a Noetherian domain, then under certain conditions we have AttR(Had+cM,N)=AttR(HacN) .

Let 𝑅 =⊕𝑛𝜖ℕ0 𝑅𝑛 be a standard graded Noetherian ring, i.e. 𝑅0 is a commutative Noetherian ring and 𝑅 is generated (as an 𝑅0 -algebra) by finitely many elements of degree one, 𝑅+ =⊕𝑛𝜖ℕ 𝑅𝑛 be the irrelevant ideal of 𝑅 and 𝒂 stands for a homogeneous ideal of 𝑅. Also, 𝑀 denotes a finitely generated graded 𝑅-module. For 𝑖𝜖ℕ0 and 𝑛𝜖ℤ let 𝐻𝒂 𝑖 (𝑀)𝑛 denotes the n-th component of the i-th graded local cohomology module 𝐻𝒂 𝑖 (𝑀) of 𝑀 with support in 𝒂 (our terminology on local cohomology comes from [3]). It is well-known that 𝐻𝑅+ 𝑖 (𝑀)𝑛 is a finitely generated 𝑅0 -module for all 𝑛𝜖ℤ and it vanishes for all sufficiently large values of n ([3, 15.1.5]). In spite of the case 𝑛 → +∞, the asymptotic behavior of 𝐻𝑅+ 𝑖 (𝑀)𝑛 when 𝑛 → −∞ is so complicated, see for example [4], and it attracts lots of interest, see [6] and [1], for a good survey on this topic. Also, in [5] the authors studied the graded components of 𝐻𝒂 𝑖 (𝑀) in the case where 𝒂 is a homogeneous ideal of 𝑅 containing the irrelevant ideal. In the case where 𝑅0 is an Artinian ring, for all 𝑖𝜖ℕ0 , ℎ𝑀 𝑖 : ℤ → ℤ(𝑛 → 𝑙𝑒𝑛𝑔𝑡ℎ𝑅0 (𝐻𝑅+ 𝑖 (𝑀)𝑛 ), is called the i-th Hilbert cohomological function of 𝑀. In [2], Brodmann et. all. studied the problem of finiteness of the set of Hilbert cohomological functions of some classes of graded modules. In this paper, we also consider this problem. Another reason of this paper, is to study the support of graded local cohomology modules 𝐻𝒂 𝑖 (𝑀) of 𝑀 with support in an arbitrary homogeneous ideal 𝒂 of 𝑅. Let cd𝒂(𝑀) ≔ sup{iϵℤ | 𝐻𝒂 𝑖 (𝑀) ≠ 0}, denotes the cohomological dimension of 𝑀 with respect to 𝒂. In [1, 3.7], it is shown that the set Supp𝑅0 (𝐻𝑅+ cd𝑅+ (𝑀) (𝑀)𝑛) is eventually stable when 𝑛 → −∞, i.e. there exists 𝑋 ⊆ Spec(𝑅0) such that Supp𝑅0 (𝐻𝑅+ 𝑐𝑑𝑅+ (𝑀) (𝑀)𝑛) = 𝑋, for all 𝑛 ≪ 0. In this paper, we study the asymptotic behavior of the set Supp𝑅0 (𝐻𝒂 𝑖 (𝑀)𝑛 ) when 𝑛 → −∞.

First, in a special case we describe 𝐻𝒂 𝑖 (𝑀) in terms of some homologies of the minimal graded free resolution of 𝑀. Then, as a consequence, we find a class of graded modules with a finite set of Hilbert cohomological functions.

We show that, in a special case, the support and dimension of 𝐻𝑅+ 𝑖 (𝑀)𝑡 have upper bound which doesn't depend on i and 𝑀 for sufficiently small values of t. Also, it is shown that, in some cases, the set {Supp𝑅0 (𝐻𝒂 𝑐𝑑𝒂(𝑀) (𝑀)𝑛)} 𝑛𝜖ℤ is eventually increasing, in the sense that Supp𝑅0 (𝐻𝒂 𝑐𝑑𝒂(𝑀) (𝑀)𝑛) ⊆ Supp𝑅0 (𝐻𝒂 𝑐𝑑𝒂(𝑀) (𝑀)𝑛−1) for all 𝑛 ≪ 0.

It is worth to find cases for the existence of a non-zero divisor x on the module M in the ideal a for which cd𝒂 (𝑀/𝑥𝑀) = cd𝒂(𝑀)− 1. It helps us to study the last non-vanishing local cohomology module of M with support in a.

Throughout this paper‎, (R, m) is a‎ commutative Noetherian local ring with the maximal ideal m. ‎The following conjecture proposed by Bass [1]‎, ‎has been‎ proved by Peskin and Szpiro [2] for almost all rings: ‎
(B) If R admits a finitely generated R-module of‎ finite injective dimension‎, ‎then R is Cohen-Macaulay.
‎The problems treated in this paper are closely related to the‎ following generalization of  Bass conjecture which is still wide‎ open:‎
(GB) If R admits a finitely generated R-module of‎ finite Gorenstein-injective dimension‎, ‎then R is‎ Cohen-Macaulay.
‎  Our idea goes back to the first steps of the solution of Bass conjecture given by Levin and Vasconcelos in 1968 [3] when R admits a‎ finitely generated R-module of injective dimension 1‎.
Levin and Vasconcelos indicate that if‎ is a non-zerodivisor‎, ‎then for‎ every finitely generated R/xR-module M‎, ‎there is‎ ‎. ‎Using this fact‎, ‎they construct a‎ finitely generated R-module of finite injective dimension in‎ the case where R is Cohen-Macaulay (the converse of Conjecture‎ B)‎.
‎ In this paper we study the Gorenstein injective dimension of local cohomology‎. ‎We also show that if R is Cohen-Macaulay‎ with minimal multiplicity‎, ‎then every finitely generated module‎ of finite Gorenstein injective dimension has finite injective‎ dimension.‎
‎We prove that a Cohen-Macaulay local ring‎ has a finitely generated module of‎ finite Gorenstein injective dimension.‎
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et  be a commutative Noetherian ring,  and  two ideals of  and  a finite -module. In this paper, we have studied the vanishing and relative Cohen-Macaulyness of the functor for relative Cohen-Macauly filtered modules with respect to the ideal  (RCMF). We have shown that the for relative Cohen-Macaulay modules holds for any relative Cohen-Macauly module with respect to  with ........