  Let $d \in \N$ and let $M$ be a finitely generated graded module of dimension
$\leq d$ over a Noetherian homogeneous ring $R$ with local Artinian base ring
$R_0$. Let $\beg(M)$, $\gendeg(M)$ and $\reg(M)$ respectively denote the
beginning, the generating degree and the Castelnuovo-Mumford regularity of $M$.
If $i \in \N_0$ and $n \in Z$, let $d^i_M(n)$ denote the $R_0$-length of the
$n$-th graded component of the $i$-th $R_+$-transform module $D^i_{R_+}(M)$ of
$M$ and let $K^i(M)$ denote the $i$-th deficiency module of $M$. Our main
result says, that $\reg(K^i(M))$ is bounded in terms of $\beg(M)$ and the
"diagonal values" $d^j_M(-j)$ with $j = 0,..., d-1$. As an application of this
we get a number of further bounding results for $\reg(K^i(M))$.
