  Let $R=\oplus_{n\in \N_0}R_n$ be a standard graded ring, $M$ be a finitely
generated graded $R$-module and $R_+:=\oplus_{n\in \N}R_n$ denotes the
irrelevant ideal of $R$. In this paper, considering the new concept of linkage
of ideals over a module, we study the graded components $H^i_{\fa}(M)_n$ when
$\fa$ is an h-linked ideal over $M$. More precisely, we show that
$H^i_{\fa}(M)$ is tame in each of the following cases: \begin{itemize}
  \item [(i)] $i=f_{\fa}^{R_+}(M)$, the first integer $i$ for which
$R_+\nsubseteq \sqrt{0:H^i_{\fa}(M)}$;
  \item [(ii)] $i=\cd(R_+,M)$, the last integer $i$ for which
$H^{i}_{R_+}(M)\neq 0$, and $\fa=\fb+R_+$ where $\fb$ is an h-linked ideal with
$R_+$ over $M$. \end{itemize} Also, among other things, we describe the
components $H^i_{\fa}(M)_n$ where $\fa$ is radically h-$M$-licci with respect
to $R_+$ of length 2.
