Indecomposablity of top local cohomology modules and connectedness of the prime divisors graphs

Let a be an ideal of a Noetherian local ring R with dim R = d and t be a positive integer. In this paper, it is shown that the top local cohomology module H-a(d)(R) (equivalently, its Matlis dual H-a(d)(R)(V)) can be written as a direct sum of t indecomposable summands if and only if the endomorphism ring Hom(R)(H-a(d)(R)(V), H-a(d)(R)(V)) can be written as a direct product of t local endomorphism rings if and only if the set of minimal primes p of R with H-a(d)(R/p) not equal 0 can be written as disjoint union of t non-empty subsets U-1, U-2, . . . , U-t such that for all distinct i, j is an element of{1,..., t} and all p is an element of U-i and all q is an element of U-j, we have ht(p+ q) >= 2. This generalizes Theorem 3.6 of Hochster and Huneke [Contemp. Math. 159 (1994) 197-208].