| Title:
|
Matlis dual of local cohomology modules (English) |
| Author:
|
Naal, Batoul |
| Author:
|
Khashyarmanesh, Kazem |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
70 |
| Issue:
|
1 |
| Year:
|
2020 |
| Pages:
|
1-7 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $(R,\mathfrak m)$ be a commutative Noetherian local ring, $\mathfrak a$ be an ideal of $R$ and $M$ a finitely generated $R$-module such that $\mathfrak a M\neq M$ and ${\rm cd}(\mathfrak a,M) - {\rm grade}(\mathfrak a,M)\leq 1$, where ${\rm cd}(\mathfrak a,M)$ is the cohomological dimension of $M$ with respect to $\mathfrak a$ and ${\rm grade}(\mathfrak a,M)$ is the $M$-grade of $\mathfrak a$. Let $D(-) := {\rm Hom}_R(-,E)$ be the Matlis dual functor, where $E := E(R/\mathfrak m)$ is the injective hull of the residue field $R/\mathfrak m$. We show that there exists the following long exact sequence \begin {eqnarray*} 0 \longrightarrow & H^{n-2}_{\mathfrak a}(D(H^{n-1}_{\mathfrak a}(M))) \longrightarrow H^{n}_{\mathfrak a}(D(H^{n}_{\mathfrak a}(M))) \longrightarrow D(M) \\ \longrightarrow & H^{n-1}_{\mathfrak a}(D(H^{n-1}_{\mathfrak a}(M))) \longrightarrow H^{n+1}_{\mathfrak a}(D(H^{n}_{\mathfrak a}(M))) \\ \longrightarrow & H^{n}_{\mathfrak a}(D(H^{n-1}_{(x_1, \ldots ,x_{n-1})}(M))) \longrightarrow H^{n}_{\mathfrak a}(D(H^{n-1}_\mathfrak (M))) \longrightarrow \ldots , \end {eqnarray*} where $n:={\rm cd}(\mathfrak a,M)$ is a non-negative integer, $x_1, \ldots ,x_{n-1}$ is a regular sequence in $\mathfrak a$ on $M$ and, for an $R$-module $L$, $H^i_{\mathfrak a}(L)$ is the $i$th local cohomology module of $L$ with respect to $\mathfrak a$. (English) |
| Keyword:
|
local cohomology module |
| Keyword:
|
Matlis dual functor, filter regular sequence |
| MSC:
|
13D07 |
| MSC:
|
13D45 |
| idZBL:
|
07217118 |
| idMR:
|
MR4078343 |
| DOI:
|
10.21136/CMJ.2019.0134-18 |
| . |
| Date available:
|
2020-03-10T10:12:26Z |
| Last updated:
|
2022-04-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148037 |
| . |
| Reference:
|
[1] Brodmann, M. P., Sharp, R. Y.: Local Cohomology. An Algebraic Introduction with Geometric Applications.Cambridge Studies in Advanced Mathematics 136, Cambridge University Press, Cambridge (2013). Zbl 1263.13014, MR 3014449, 10.1017/CBO9781139044059 |
| Reference:
|
[2] Hellus, M.: On the associated primes of Matlis duals of top local cohomology modules.Commun. Algebra 33 (2005), 3997-4009. Zbl 1101.13026, MR 2183976, 10.1080/00927870500261314 |
| Reference:
|
[3] Hellus, M.: Local Cohomology and Matlis Duality.Habilitationsschrift, Leipzing Available at https://www.uni-regensburg.de/mathematik/mathematik-hellus/ medien/habilitationsschriftohnedeckblatt.pdf (2006). |
| Reference:
|
[4] Hellus, M.: Finiteness properties of duals of local cohomology modules.Commun. Algebra 35 (2007), 3590-3602. Zbl 1129.13018, MR 2362672, 10.1080/00927870701512069 |
| Reference:
|
[5] Hellus, M., Schenzel, P.: Notes on local cohomology and duality.J. Algebra 401 (2014), 48-61. Zbl 1304.13033, MR 3151247, 10.1016/j.jalgebra.2013.12.006 |
| Reference:
|
[6] Khashyarmanesh, K.: On the finiteness properties of extension and torsion functors of local cohomology modules.Proc. Am. Math. Soc. 135 (2007), 1319-1327. Zbl 1111.13016, MR 2276640, 10.1090/s0002-9939-06-08664-3 |
| Reference:
|
[7] Khashyarmanesh, K.: On the Matlis duals of local cohomology modules.Arch. Math. 88 (2007), 413-418. Zbl 1112.13020, MR 2316886, 10.1007/s00013-006-1115-1 |
| Reference:
|
[8] Khashyarmanesh, K., Salarian, S.: Filter regular sequences and the finiteness of local cohomology modules.Commun. Algebra 26 (1998), 2483-2490. Zbl 0909.13007, MR 1627876, 10.1080/00927879808826293 |
| Reference:
|
[9] Khashyarmanesh, K., Salarian, S.: On the associated primes of local cohomology modules.Commun. Algebra 27 (1999), 6191-6198. Zbl 0940.13013, MR 1726302, 10.1080/00927879908826816 |
| Reference:
|
[10] Schenzel, P.: Matlis duals of local cohomology modules and their endomorphism rings.Arch. Math. 95 (2010), 115-123. Zbl 1200.13028, MR 2674247, 10.1007/s00013-010-0149-6 |
| Reference:
|
[11] Schenzel, P., Trung, N. V., Cuong, N. T.: Verallgemeinerte Cohen-Macaulay-Moduln.Math. Nachr. 85 (1978), 57-73 German. Zbl 0398.13014, MR 0517641, 10.1002/mana.19780850106 |
| Reference:
|
[12] Stückrad, J., Vogel, W.: Buchsbaum Rings and Applications. An Interaction Between Algebra, Geometry and Topology.Springer, Berlin (1986). Zbl 0606.13018, MR 0881220 |
| . |
