Hilbert-Samuel coefficients and postulation numbers of graded components of certain local cohomology modules

Abstract

Let $R=\bigoplus_{n\ge 0}R_n$ be a positively graded commutative Noetherian ring with one-dimensional local base ring $(R_0,\germ{m}_0)$. Let $\germ{q}_0\subseteq R_0$ be an $\germ{m}_0$-primary ideal. Let $M$ be a finitely generated graded $R$-module and let $i\in\Bbb N_0$. Let $H^i_{R_+}(M)$ denote the $i$-th local cohomology module of $M$ with respect to the irrelevant ideal $R_+=\bigoplus_{n>0}R_n$ of $R$. The main results in this paper are: \roster \item"(a)" Assume that $\dim_{R_0}(H^i_{R_+}(M)_n)=1$ for all $n\ll 0$. Then there is a polynomial $S(x)\in\Bbb Q[x]$ of degree $<i$ such that for all $n\ll 0$, $$e_1(\germ{q}_0,H^i_{R_+}(M)_n)=S(n),$$ where $e_1(\germ{q}_0,H^i_{R_+}(M)_n)$ is the so-called first Hilbert-Samuel coefficient of $H^i_{R_+}(M)_n$ with respect to $\germ{q}_0$. \item"(b)" There is some $c\in\Bbb N_0$ such that $\mu(\germ{q}_0,H^i_{R_+}(M)_n)\le c$ for all $n\in\Bbb Z$, where $\mu(\germ{q}_0,H^i_{R_+}(M)_n)$ is the postulation number of the $R_0$-module $H^i_{R_+}(M)_n$ with respect to $\germ{q}_0$. \endroster