  We study the module of Koszul cycles $Z_t(I,M)$ of a homogeneous ideal $I$ in
a polynomial ring $S$ with respect to a graded module $M$. Under mild
assumptions on the base field we prove that the regularity of $Z_t(I,S)$ is a
subadditive function of the homological position t when I is 0-dimensional. For
Borel-fixed ideals $I$ and $J$ we prove that the regularity of $Z_t(I,S/J)$ is
bounded above by $t(1+\reg I)+\reg S/J$.
