Cohen-Macaulay filtered modules and attached primes of local cohomology

A. Atazadehr, R. Naghipour

For an ideal a in a Noetherian ring R contained in the Jacobson radical of R, it is shown that if M is a finitely generated a-relative Cohen-Macaulay R-module, then AnnR(Hacd(a, M)(M)) = AnnR(M). As an application of this result, we show that if M is a finitely generated a-relative Cohen-Macaulay filtered R-module with the cohomological dimension filtration M ={Mi}0≤i≤c, then for each 0≤i≤c, AnnR(Hai(M)) = AnnR(Mi / Mi-1), where c=cd(a, M). These generalize the main results of [9, Theorem 3.3] and [5, Theorem 2.11]. Also, we shall provide some new characterizations of the attached primes of top local cohomology module Hacd(a, M)(M) and give a short proof of the main results of [1, Theorem 2.2] and [13, Theorem 2.7]. Finally, it is shown that if M and N are arbitrary R-modules (not necessarily finitely generated) such that AttR(M) ⊆ AttR(N), then cd(a, R/AnnR(M)) ≤ cd(a, R/AnnR(N)). 

Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 17(3) (2024), pp. 39-49