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

Let $R = \bigoplus _{n \geq 0} R_n$ be a Noetherian homogeneous ring with one-dimensional local base ring $(R_0, {\mathfrak m}_0)$. Let ${\mathfrak q}_0 \subseteq R_0$ be an ${\mathfrak m}_0$-primary ideal, let $M$ be a finitely generated graded $R$-module and let $i \in {\mathbb N}_0$. 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 $R$. We show that the first Hilbert-Samuel coefficient $e_1 \big( {\mathfrak q}_0, H^i_{R_+}(M)_n \big)$ of the $n$-th graded component of $H^i_{R_+}(M)$ with respect to ${\mathfrak q}_0$ is antipolynomial of degree $< i$ in $n$. In addition, we prove that the postulation numbers of the components $H^i_{R_+} (M)_n$ with respect to ${\mathfrak q}_0$ have a common upper bound.