Let and let M be a finitely generated graded module of dimension 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 and nZ, let denote the R
0-length of the n-th graded component of the i-th R
+-transform module of M and let K
i(M) denote the i-th deficiency module of M.
Our main result says, that reg(Ki(M)) is bounded in terms of beg(M) and the “diagonal values” with j=0,…,d−1. As an application of this we get a number of further bounding results for reg(Ki(M)).