Filter depth of complexes

Abstract

Let $\mathfrak{a},\mathfrak{b}$ be two ideals of a commutative noetherian ring R. The concept of a $\mathfrak{b}$-filter depth of $\mathfrak{a}$ on an arbitrary R-complex M, $\mathrm{f}-\text{depth}(\mathfrak{b},\mathfrak{a},M),$ is introduced and several characterizations via $\text{RHom},$ Koszul complexes and local cohomology are given. Some bounds of $\mathrm{f}-\text{depth}(\mathfrak{b},\mathfrak{a},M)$ for an R-complex $0\cancel{\simeq }M\in {\mathrm{D}}_{\mathrm{f}}^{+}(R)$ such that ${\text{Supp}}_R(R/\mathfrak{a}{\otimes }_R^{\mathrm{L}}M)⊈\mathcal{V}(\mathfrak{b})$ are provided, in special cases, recover and generalize the known results about the usual (filter-) depth of modules.

Keywords:

• Filter depth
• local cohomology

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13D02
• 13D45
• 13E05

Acknowledgments

We sincerely thank the referee for his or her valuable suggestions and comments, and for pointing out some errors in the manuscript.

Additional information

Funding

This research was partially supported by National Natural Science Foundation of China (11761060, 11901463), Improvement of Young Teachers’ Scientific Research Ability (NWNU-LKQN-18-30), and Innovation Ability Enhancement Project of Gansu Higher Education Institutions (2019A-002).