Attached primes of top local cohomology modules

 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) .