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Hilbert-Kirby Polynomials in Generalized Local Cohomology Modules
M. Shafiei
,
A. Khojali
,
A. Azari
,
N. Zamani
Abstract
Let $R=\oplus_{n\in\mathbb{N}_0}R_n$ be a Noetherian homogeneous ring with irrelevant ideal $R_+=\oplus_{n\in\mathbb{N}}R_n$ and with local base ring $(R_0,\mathfrak{m}_0)$. Let $M$, $N$ be two finitely generated $\mathbb{Z}$-graded $R$-modules. We show that the lengths of the graded components of various graded submodules and quotients of the $i$-th generalized local cohomology $H^i_{R_+}(M, N)$ are anti-polynomial. Under some mild assumptions, the Artinianness of $H^i_{R_+}(M, N)$ and the asymptotic behaviour of the $R_0$-modules $H^i_{R_+}(M, N)_n$ for $n\rightarrow -\infty$ in the range $i\leq\inf\{i\in \mathbb{N}_{0} \vert \sharp\{n\vert\ell_{R_{0}}(H^i _{ R_+}(M , N)_n)$ $ = \infty\}=\infty\}$ will be studied. Moreover, it has been proved that, if $u$ is the least integer $i$ for which $H^i_{R_+}(M,N)$ is not Artinian and $\mathfrak q_0$ is an $\mathfrak m_0$-primary ideal of $R_0$, then $H^u_{R_+}(M,N)/\mathfrak q_0H^u_{R_+}(M,$ $N)$ is Artinian with Hilbert-Kirby polynomial of degree less than $u$. In particular, with $M=R$, we deduce the correspondence result for ordinary local cohomology module $H^i_{R_+}(N)$.