On the top local cohomology modules

Let (R,m)(R,m) be a Noetherian local ring and I an ideal of R. Let M be a finitely generated R  -module with dimM=d. It is clear by Matlis duality that if R   is complete then HId(M) satisfies the following property:equation(⁎)AnnR(0:HId(M)p)=pfor all prime ideals p⊇AnnRHId(M). However, HId(M) does not satisfy the property (⁎) in general. In this paper we characterize the property (⁎) of HId(M) in order to study the catenarity of the ring R/AnnRHId(M), the set of attached primes AttRHId(M), the co-support CosR(HId(M)), and the multiplicity of HId(M). We also show that if HId(M) satisfies the property (⁎) then HId(M)≅Hmd(M/N) for some submodule N of M.