On the pertubations of E-depth, Ef-depth and sequentially Cohen-Macaulay locus

Abstract

Let (R,m) be a Noetherian local ring, I an ideal of R and M a finitely generated R-module. In this paper, we define and study E-depth and Ef-depth of M in I. We prove that E-depth (resp. Ef-depth under mild conditions) of M in I is the common length of all maximal sequential sequences (resp. maximal sequential f-sequences) of M in I. We show that E-depth and Ef-depth do not decrease under small pertubations. We describe the non sequentially Cohen-Macaulay locus nSCM(M) of M in terms of support and non Cohen-Macaulay locus of deficiency modules of M. Finally, we show that the dimension of non sequentially Cohen-Macaulay locus with respect to a sequential f-sequence does not increase under small pertubations.