  We generalize a result of Eisenbud-Huneke-Ulrich on the maximal graded shifts
of a module with prescribed annihilator and prove a linear regularity bound for
ideals in a polynomial ring depending only on the first $p - c$ steps in the
resolution, where $p = \mathrm{pd}(S/I)$ and $c = \mathrm{codim}(I)$.
