ایده‌آل‌های اول چسبیدۀ‌ آخرین مدول‌ ناصفرکوهمولوژی موضعی

Introduction

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 by

HaiM,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 is

supi∈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.

Material and methods

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.



Results and discussion

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).

Conclusion

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) .