Local cohomology over homogeneous rings with one-dimensional local base ring

Abstract:

Let $R=\bigoplus _{n\geq 0}R_n$ be a homogeneous Noetherian ring with local base ring $(R_0,\mathfrak {m}_0)$ and let $M$ be a finitely generated graded $R$-module. Let $H^i_{R_+}(M)$ be the $i$-th local cohomology module of $M$ with respect to $R_+:=\bigoplus _{n>0}R_n$. If $\dim R_0\leq 1$, the $R$-modules $\Gamma _{\mathfrak {m}_0R}(H^i_{R_+}(M))$, $(0:_{H_{R_+}^i(M)}\mathfrak {m}_0)$ and $H^i_{R_+}(M)/\mathfrak {m}_0H^i_{R_+}(M)$ are Artinian for all $i\in \mathbb {N}_0$. As a consequence, much can be said on the asymptotic behaviour of the $R_0$-modules $H^i_{R_+}(M)_n$ for $n\rightarrow -\infty$.