Regularity bounds for Koszul cycles

Abstract:

We study the Castelnuovo-Mumford regularity of 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 $M$ in the homological position $t\in {\mathbb {N}}$. Under mild assumptions on the base field we prove that $\operatorname {reg} Z_t(I,S)$ is a subadditive function of $t$ when $\operatorname {dim} S/I=0$. For Borel-fixed ideals $I,J$ we prove that $\operatorname {reg} Z_t(I,S/J)\leq t(1+ \operatorname {reg} I)+\operatorname {reg} S/J$, a result already announced by Bruns, Conca and Römer.