Associated and attached primes of local cohomology modules over almost Dedekind domains

Abstract

In this paper, we prove the Grothendieck’s Vanishing Theorem for finitely generated ideals over almost Dedekind domains and show that there exists a non-finitely generated prime ideal over a non-Noetherian almost Dedekind domain that does not satisfy the Grothendieck’s Vanishing Theorem. Among the other results, by considering the attached prime ideals and associated prime ideals of representable modules over almost Dedekind domains, we show that if M is a representable R-module and for every non-zero submodule N of M, the ring has acc on d-annihilators, then . Also, if M is representable and for every non-zero submodule N of , the ring has acc on d-annihilators, then , where is an injective envelope of M.