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

Abstract

Let [Formula: see text] be an ideal of a Noetherian local ring [Formula: see text] with [Formula: see text] and [Formula: see text] be a positive integer. In this paper, it is shown that the top local cohomology module [Formula: see text] (equivalently, its Matlis dual [Formula: see text]) can be written as a direct sum of [Formula: see text] indecomposable summands if and only if the endomorphism ring [Formula: see text] can be written as a direct product of [Formula: see text] local endomorphism rings if and only if the set of minimal primes [Formula: see text] of [Formula: see text] with [Formula: see text] can be written as disjoint union of [Formula: see text] non-empty subsets [Formula: see text] such that for all distinct [Formula: see text] and all [Formula: see text] and all [Formula: see text], we have [Formula: see text]. This generalizes Theorem 3.6 of Hochster and Huneke [Contemp. Math. 159 (1994) 197–208].