| Title:
|
Some results on cohomology modules with respect to an ideal and a non-negative integer (English) |
| Author:
|
Sayedsadeghi, Mirsadegh |
| Author:
|
Masoudiporlir, Seyedhamid |
| Author:
|
Sadeghi, Miryousef |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0011-4642 |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
151 |
| Issue:
|
3 |
| Year:
|
2026 |
| Pages:
|
357-375 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $R$ be a commutative Noetherian ring, $J$ an ideal of $R$, $M$ an $R$-module and let $d$ be a non-negative integer. In this paper we introduce a generalization of the notion of a local cohomology module, which we call a local cohomology module with respect to non-negative integer $d$ and ideal $J$, and study some of its properties. Also, we define $W(d,J)$ and $\widetilde {W}(d,J)$ and present the conditions under which $H^i_{d,J}(M)=0$. Finally, we investigate the concept of ${\rm depth}_{d,J}(M)$ and examine the top vanishing of the $R$-module $H^i_{d,J}(M)$. (English) |
| Keyword:
|
$(d,J)$-local cohomology |
| Keyword:
|
system of ideals |
| Keyword:
|
$(d,J)$-depth, $W(d,J)$ |
| MSC:
|
13D45 |
| MSC:
|
13E05 |
| MSC:
|
14B15 |
| DOI:
|
10.21136/MB.2025.0110-24 |
| . |
| Date available:
|
2026-08-24T07:49:57Z |
| Last updated:
|
2026-08-24 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153711 |
| . |
| Reference:
|
[1] Ahmadi-Amoli, K., Sadeghi, M. Y.: On the local cohomology modules defined by a pair of ideals and Serre subcategory.J. Math. Ext. 7 (2013), 47-62. Zbl 1327.13056, MR 3248761 |
| Reference:
|
[2] Banica, C., Stoia, M.: Singular sets of a module on local cohomology.Boll. Unione Mat. Ital., V. Ser., B 16 (1979), 923-934. Zbl 0468.13010, MR 0553806 |
| Reference:
|
[3] Bass, H.: On the ubiquity of Gorenstein rings.Math. Z. 82 (1963), 8-28. Zbl 0112.26604, MR 0153708, 10.1007/BF01112819 |
| Reference:
|
[4] Brodmann, N. P., Sharp, R. Y.: Local Cohomology: An Algebraic Introduction with Geometric Applications.Cambridge Studies in Advanced Mathematics 60. Cambridge University Press, Cambridge (1998). Zbl 0903.13006, MR 1613627, 10.1017/CBO9780511629204 |
| Reference:
|
[5] Bruns, W., Herzog, J.: Cohen-Macaulay Rings.Cambridge Studies in Advanced Mathematics 39. Cambridge University Press, Cambridge (1993). Zbl 0788.13005, MR 1251956, 10.1017/CBO9780511608681 |
| Reference:
|
[6] Chu, L., Wang, Q.: Some results on local cohomology defined by a pair of ideals.J. Math. Kyoto Univ. 49 (2009), 193-200. Zbl 1174.13024, MR 2531134, 10.1215/kjm/1248983036 |
| Reference:
|
[7] Matsumura, H.: Commutative Ring Theory.Cambridge Studies in Advanced Mathematics 8. Cambridge University Press, Cambridge (1986). Zbl 0603.13001, MR 0879273, 10.1017/CBO9781139171762 |
| Reference:
|
[8] Rotman, J. J.: Introduction to Homological Algebra.Universitext. Springer, Berlin (2009). Zbl 1157.18001, MR 2455920, 10.1007/978-0-387-68324-9 |
| Reference:
|
[9] Takahashi, R., Yoshino, Y., Yoshizawa, T.: Local cohomology based on a nonclosed-support defined by a pair of ideals.J. Pure. Appl. Algebra 213 (2009), 582-600. Zbl 1160.13013, MR 2483839, 10.1016/j.jpaa.2008.09.008 |
| Reference:
|
[10] Zamani, N.: Generalized local cohomology relative to $(I,J)$.Southeast Asian Bull. Math. 35 (2011), 1045-1050. Zbl 1265.13020, MR 2907771 |
| Reference:
|
[11] Zamani, N., Bijan-Zadeh, M. H., Sayedsadeghi, M. S.: $d$-transform functor and some finiteness and isomorphism results.Vietnam J. Math. 42 (2014), 179-186. Zbl 1308.13030, MR 3218854, 10.1007/s10013-013-0042-2 |
| Reference:
|
[12] Zamani, N., Bijan-Zadeh, M. H., Sayedsadeghi, M. S.: Cohomology with supports of dimension $\leq d$.J. Algebra Appl. 15 (2016), Article ID 1650042, 10 pages. Zbl 1336.13008, MR 3454704, 10.1142/S0219498816500420 |
| . |
