On Depth of an Ideal on Artinian Modules and Applications
Carlos Henrique Tognon *
Department of Mathematics, ICTE, Federal University of Triangulo Mineiro, Uberaba, MG, Brazil.
*Author to whom correspondence should be addressed.
Abstract
Let (R,m) be a commutative Noetherian local ring, let M be an Artinian ZD-module, and let S be a Serre subcategory of the category of R-modules satisfying condition Cm. This study examines S-sequences and the associated notion of S-depth in the Artinian setting. Under the stated hypotheses, it is shown that, whenever m contains a maximal S-sequence on M, all maximal S-sequences in m have the same length. This common length is characterised through the least index at which the relevant Ext-module or local cohomology module does not belong to S.
The behaviour of S-depth is also considered in short exact sequences of Artinian ZD-modules, yielding inequalities that relate the corresponding depth values of the constituent modules. A further connection is developed between depth and Bass numbers through localisation and Ext-based descriptions.
The theoretical discussion is supplemented by examples intended to illustrate the influence of the chosen Serre subcategory on S-regularity, the existence of maximal S-sequences, and the computation of S-depth. Overall, the work presents a homological framework for studying generalised depth invariants in Artinian modules and clarifies how Serre-subcategory conditions interact with local cohomology, Ext-functors, exact sequences, and Bass-number formulations. The findings are formulated as structural results and provide a basis for further examination of generalised depth beyond the finitely generated setting.
Several special cases follow from the theorem, including results for quasi-\(\delta\)-power increasing sequences, | \(\bar{N}\), pn|k summability, and |B, pn|k summability under appropriate choices of parameters. Thus, the theorem provides a unified formulation for related absolute summability factor results within the stated hypotheses. No additional assumptions are introduced beyond those specified in the theorem, and the derived consequences are presented only as formal reductions of the main result. This maintains a close connection between the generalised theorem and the earlier summability factor results considered in the manuscript.
Keywords: Artinian module, ZD-module, serre subcategory, S-sequence, S-depth, local cohomology, ext-module, Bass number, Noetherian local ring, homological invariant