Journal of Advanced Mathematical Modeling

Journal of Advanced Mathematical Modeling

on the lengths and multiplicities of some graded generalized local cohomology modules

Document Type : Original Paper

Authors
Saeedeh Karimbeyglu
ahmad khojali
Naser Zamani
Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran
10.22055/jamm.2025.48234.2325
Abstract
Let $R=\oplus _{n\in\mathbb{N}_0}R_n$ be a Noetherian homogeneous ring with a local base ring $(R_0,\mathfrak m_0)$ of Krull dimension 2. Let $R_+=\oplus _{n\in\mathbb{N}}R_n$ be the irrelevant ideal of $R$. Let $M$ and $N$ be two finitely generated $\mathbb{Z}$-graded $R$-modules such that, $\textrm{pd}_R(M)$, the projective dimension of $M$, is finite. Under some conditions on $H^i_{R_+}\!(M,N)$, it will be shown that there exists $x\in\frak m_0$ such that $\ell_{R_0}((0:_{H^i_{R_+}\!(M,N)_n}\!\!x))$ is of reverse polynomial type of degree less than $i$; i.e., there exists a polynomial $P(X)\in\mathbb{Q}[X]$ of degree less than $i$ such that
$\ell_{R_0}((0:_{H^i_{R_+}\!(M,N)_n}\!\!x))=P(n)$, for all $n\ll 0$. Also, it is proved that, $\ell_{R_0}(\Gamma_{\!\frak m_0R}H^i_{R_+}\!(M, N/xN)_n)$ and $\ell_{R_0}(H^1_{\!\frak m_0R}H^i_{R_+}\!(M, N)_n)$, for a parameter element $x\in \frak m_0$, are of reverse polynomial type of degree less than $i$. Finally, for an $\frak m_0$-primary ideal $\frak q_0$, we study the asymptotic behaviour of the multiplicities $e_d(\frak q_0, H^i_{R_+}(M,N)_n)$ for $n\ll 0$, where $d=\dim_{\widehat{R}}\big(^*\!D(H^{i}_{R_+}\!(M, N)\otimes_{R_0}\widehat{R_0})\big)$.
Keywords
Filter regular element
generalized local cohomology
Hilbert-Kirby polynomial
irrelevant ideal
Multiplicity
Subjects
[1] Brodmann, M., Fumasoli, S. and Tajarod, R., 2003. Local cohomology over homogeneous rings with
one-dimensional local base ring. Proc. Amer. Math. Soc., 131(10), pp.2977-2985. doi: 10.1090/S0002-
9939-03-07009-6
[2] Brodmann, M. and Hellus, M., 2002. Cohomological patterns of coherent sheaves over projective
schemes. J. Pure Appl. Algebra, 172(2-3), pp.165-182. doi: 10.1016/S0022-4049(01)00144-X
[3] Brodmann, M., Rohrer, F. and Sazeedeh, R., 2005. Multiplicities of graded components of local
cohomology modules. J. Pure Appl. Algebra, 197(1-3), pp.249-278. doi: 10.1016/j.jpaa.2004.08.034
[4] Brodmann, M. and Sharp, R. Y., 2013. Local cohomology: an algebraic introduction with geometric
applications (Vol. 136). Cambridge university press.
[5] Bruns, W. and Herzog, J., 1993. Cohen-macaulay rings (No. 39). Cambridge university press.
[6] Lech, C., 1957. On the associativity formula for multiplicities. Ark. Mat., 3(4), pp.301-314. doi:
10.1007/BF02589424
[7] Cuong, N. T. and Van Hoang, N., 2005. Some finite properties of generalized local cohomology
modules. East West Math., 7(02), pp.107-115. doi: 10.645-1-10-20200302
[8] Cuong, N. T. and Van Hoang, N., 2008. On the vanishing and the finiteness of supports of generalized
local cohomology modules. Manuscripta Math., 126(1), pp.59-72. doi: 10.1007/s00229-007-0162-7
[9] Divaani-Aazar, K. and Hajikarimi, A., 2011. Generalized local cohomology modules and homological
Gorenstein dimensions. Comm. Alg., 39(6), pp.2051-2067. doi: 10.1080/00927870903295380
[10] Enochs, E. E. and Jenda, O. M., 2011. Relative homological algebra: Volume 1 (Vol. 30). Walter de
Gruyter.
[11] Herzog, J., 1974. Komplexe, Aufl¨osungen und Dualit¨at in der lokalen Algebra. Habilitationsschrift,
Universit¨at Regensburg.
[12] Jahangiri, M., Shirmohammadi, N. and Tahamtan, S., 2015. Tameness and Artinianness of
Graded Generalized Local Cohomology Modules. Algebra Colloq. Vol. 22, No. 01, pp. 131-146. doi:
10.1142/S1005386715000127
[13] D. Kirby Kirby, D., 1973. Artinian modules and Hilbert polynomials. Q. J. Math., 24(1), pp.47-57.
doi: 10.1093/qmath/24.1.47
[14] Shafiei, M., Khojali, A., Azari, A. and Zamani, N., 2021. Hilbert-Kirby Polynomials in Generalized
Local Cohomology Modules. Acta Math. Vietnam., 46, pp.747-759. doi: 10.1007/s40306-021-00440-3
[15] Rotthaus, C. and S¸ega, L., 2006. Open loci of graded modules. Trans. Amer. Math. Soc., 358(11),
pp.4959-4980. doi: 10.1090/s0002-9947-06-03876-1
[16] Sharp, R. Y. and V´amos, P., 1985. Baire’s category theorem and prime avoidance in complete local
rings. Arch. Math., 44, pp.243-248. doi: 10.1007/BF01237858
[17] Suzuki, N., 1978. On the generalized local cohomology and its duality. J. Math. Kyoto Univ., 18(1),
pp.71-85. doi: 10.1215/kjm/1250522630
ON THE LENGTHS AND MUL
18] Zamani, N., 2003. On the homogeneous pieces of graded generalized local cohomology modules.
Colloq. Math. (Vol. 97, No. 2, pp. 181-188). doi:10.4064/cm97-2-5
[19] Zamani, N., 2006. On graded generalized local cohomology. Arch. Math., 86(4), pp.321-330. doi:
10.1007/s00013-005-1524-6
[20] Zamani, N. and Khojali, A., 2015. Artinian graded generalized local cohomology. J. Algebra Appl.,
14(07), p.1550111. doi: 10.1142/S021949881550111X 

  • Receive Date 29 October 2024
  • Revise Date 01 March 2025
  • Accept Date 25 September 2025