Depth of an ideal on ZD-modules

摘要

Let R be a Noetherian ring, I an ideal of R and M a ZD-module. Let S be a Serre subcategory of the category of R-modules satisfying the condition CI, and let I contain a maximal S-sequence on M. We show that all maximal S-sequences on M in I, have the same length. If this common length is denoted by S-depth(I,M), then S-depth(I,M) = inf{i:ExtiR(R/I,M) ?S} = inf{i:HiI (M)?S}. Also some properties of this notion are investigated. It is proved that S-depth(I,M) = inf{depthMp : p ?V (I) and R/p ? S} = inf{S-depth(p,M) : p ? V (I) and R/p ? S} whenever S is a Serre subcategory closed under taking injective hulls, and M is a ZD-module.