Some characterizations of relative sequentially Cohen–Macaulay and relative Cohen–Macaulay modules

Abstract

Let [Formula: see text] be an [Formula: see text]-module over a Noetherian ring [Formula: see text] and [Formula: see text] an ideal of [Formula: see text] with c = [Formula: see text] ([Formula: see text], R). First, over an [Formula: see text]-relative Cohen–Macaulay local ring [Formula: see text], we provide a characterization of the [Formula: see text]-relative sequentially Cohen–Macaulay modules [Formula: see text] in terms of [Formula: see text]-relative Cohen–Macaulayness of the [Formula: see text]-modules [Formula: see text] for all [Formula: see text], where [Formula: see text]. Next, we prove that [Formula: see text] is finite [Formula: see text]-relative Cohen–Macaulay if and only if [Formula: see text] for all [Formula: see text] and [Formula: see text]. Finally, we provide another characterization of [Formula: see text]-relative sequentially Cohen–Macaulay modules [Formula: see text] in terms of vanishing of the local homology modules [Formula: see text] for all [Formula: see text] and for all [Formula: see text].