مجله مدل‌سازی پیشرفته ریاضی

مجله مدل‌سازی پیشرفته ریاضی

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

نوع مقاله : مقاله پژوهشی

نویسندگان
Saeedeh Karimbeyglu
Ahmad Khojali
Naser Zamani
Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran
10.22055/jamm.2025.48234.2325
چکیده
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 $d=\dim_{\widehat{R}}\big(^*\!D(H^{i}_{R_+}\!(M, N)\otimes_{R_0}\widehat{R_0})\big)$.
کلیدواژه‌ها
Filter regular element
generalized local cohomology
Hilbert-Kirby polynomial
irrelevant ideal
Multiplicity
موضوعات

عنوان مقاله English

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

نویسندگان English

Saeedeh Karimbeyglu
ahmad khojali
Naser Zamani
Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran
چکیده English

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 $d=\dim_{\widehat{R}}\big(^*\!D(H^{i}_{R_+}\!(M, N)\otimes_{R_0}\widehat{R_0})\big)$.

کلیدواژه‌ها English

Filter regular element
generalized local cohomology
Hilbert-Kirby polynomial
irrelevant ideal
Multiplicity
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ON THE LENGTHS AND MUL
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  • تاریخ دریافت 08 آبان 1403
  • تاریخ بازنگری 11 اسفند 1403
  • تاریخ پذیرش 03 مهر 1404