Artinianess of graded local cohomology modules

Abstract:

Let $R=\bigoplus _{n \in \mathbb {N}} R_n$ be a Noetherian homogeneous ring with local base ring $(R_0, \mathfrak {m}_0)$ and let $M$ be a finitely generated graded $R$-module. Let $a$ be the largest integer such that $H_{R_+}^a(M)$ is not Artinian. We will prove that $H_{R_+}^i(M)/\mathfrak {m}_0H_{R_+}^i(M)$ are Artinian for all $i\geq a$ and there exists a polynomial $\widetilde {P}\in \mathbb {Q}[\mathbf {x}]$ of degree less than $a$ such that $\textrm {length}_{R_0}(H_{R_+}^a(M)_n /\mathfrak {m}_0H_{R_+}^a(M)_n) =\widetilde {P}(n)$ for all $n\ll 0$. Let $s$ be the first integer such that the local cohomology module $H_{R_+}^s(M)$ is not ${R_+}-$cofinite. We will show that for all $i\leq s$ the graded module $\Gamma _{\mathfrak {m}_0}(H_{R_+}^i(M))$ is Artinian.