Let $ S $ be a commutative Noetherian ring and $ \mathfrak{a} $ a proper ideal of $ S $. We give several bounds of $ \mathfrak{a} $-depth of $ S $-complexes and $ S $-modules, investigate the behavior of $ \mathfrak{a} $-depth and $ \mathfrak{a} $-Cohen-Macaulay modules under tensor product with a faithfully flat $ S $-module. Furthermore, we establish the Foxby equivalence of $ \mathfrak{a} $-Cohen-Macaulay $ S $-modules.
Citation: Pengju Ma, Yanjie Li. $ \mathfrak{a} $-Depth and $ \mathfrak{a} $-Cohen-Macaulay modules[J]. Electronic Research Archive, 2026, 34(2): 1195-1208. doi: 10.3934/era.2026055
Let $ S $ be a commutative Noetherian ring and $ \mathfrak{a} $ a proper ideal of $ S $. We give several bounds of $ \mathfrak{a} $-depth of $ S $-complexes and $ S $-modules, investigate the behavior of $ \mathfrak{a} $-depth and $ \mathfrak{a} $-Cohen-Macaulay modules under tensor product with a faithfully flat $ S $-module. Furthermore, we establish the Foxby equivalence of $ \mathfrak{a} $-Cohen-Macaulay $ S $-modules.
| [1] | W. Bruns, H. J. Herzog, Cohen-Macaulay Rings, $2^{nd}$ edition, Cambridge University Press, 2009. https://doi.org/10.1017/CBO9780511608681 |
| [2] | E. Enochs, S. Yassemi, Foxby equivalence and cotorsion theories relative to semidualizing modules, Math. Scand., 95 (2004), 33–43. |
| [3] |
K. A. Beigi, K. Divaani-Aazar, M. Tousi, On the invariance of certain types of generalized Cohen-Macaulay modules under Foxby equivalence, Czech. Math. J., 72 (2022), 989–1002. https://doi.org/10.21136/CMJ.2022.0227-21 doi: 10.21136/CMJ.2022.0227-21
|
| [4] |
H. Holm, D. White, Foxby equivalence over associative rings, J. Math. Kyoto Univ., 47 (2007), 781–808. https://doi.org/10.1215/kjm/1250692289 doi: 10.1215/kjm/1250692289
|
| [5] |
Z. Zhang, J. Wei, Relative copure modules and Foxby equivalence, Commun. Algebra, 49 (2021), 3223–3231. https://doi.org/10.1080/00927872.2021.1892710 doi: 10.1080/00927872.2021.1892710
|
| [6] |
R. Zhao, Y. Li, A unified approach to various Gorenstein modules, Rocky Mountain J. Math., 52 (2022), 2229–2246. https://doi.org/10.1216/rmj.2022.52.2229 doi: 10.1216/rmj.2022.52.2229
|
| [7] |
N. X. Linh, L. T. Nhan, On sequentially Cohen-Macaulay modules and sequentially generalized Cohen-Macaulay modules, J. Algebra, 678 (2025), 635–653. https://doi.org/10.1016/j.jalgebra.2025.04.024 doi: 10.1016/j.jalgebra.2025.04.024
|
| [8] | P. Schenzel, On the dimension filtration and Cohen-Macaulay filtered modules, Lect. Notes Pure Appl. Math., (1999), 245–264. |
| [9] | J. Stückrad, W. Vogel, Buchsbaum Rings and Applications: An Interaction Between Algebra, Geometry and Topology, $1^{st}$ edition, Springer, 1986. |
| [10] | W. Mahmood, M. Azam, $I$-Cohen Macaulay modules, preprint, arXiv: 1906.00143. |
| [11] |
H. Foxby, Bounded complexes of flat modules, J. Pure Appl. Algebra, 15 (1979), 149–172. https://doi.org/10.1016/0022-4049(79)90030-6 doi: 10.1016/0022-4049(79)90030-6
|
| [12] | S. B. Iyengar, G. J. Leuschke, A. Leykin, C. Miller, E. Miller, A. K. Singh, et al., Twenty-Four Hours of Local Cohomology, American Mathematical Society, 2007. |
| [13] | L. W. Christensen, H. Foxby, Hyperhomological algebra with applications to commutative Rings, 2006. Available from: https://www.math.ttu.edu/lchriste/download/918-all.pdf. |
| [14] | L. W. Christensen, H. Foxby, H. Holm, Derived Category Methods in Commutative Algebra, $1^{st}$ edition, Springer, 2024. https://doi.org/10.1007/978-3-031-77453-9 |
| [15] |
S. Iyengar, Depth for complexes, and intersection theorem, Math. Z., 230 (1999), 545–567. https://doi.org/10.1007/PL00004705 doi: 10.1007/PL00004705
|
| [16] | B. Singh, Basic Commutative Algebra, World Scientific, 2011. https://doi.org/10.1142/7811 |
| [17] | M. F. Atiyah, I. G. Macdonald, Introduction To Commutative Algebra, $1^{st}$ edition, CRC Press, 1969. https://doi.org/10.1201/9780429493638 |
| [18] | S. Sather-Wagstaff, Semidualizing Modules, 2010. Available from: https://ssather.people.clemson.edu/DOCS/sdm.pdf. |
| [19] | L. W. Christensen, Semi-dualizing complexes and their Auslander categories, Trans. Am. Math. Soc., 353 (2001), 1839–1883. |
| [20] | N. Bourbaki, Commutative Algebra: Chapters 1–7, $1^{st}$ edition Springer, 1998. |
Pengju Ma, Yanjie Li. $ \mathfrak{a} $-Depth and $ \mathfrak{a} $-Cohen-Macaulay modules[J]. Electronic Research Archive, 2026, 34(2): 1195-1208. doi: 10.3934/era.2026055
DownLoad: