Koszul cycles

Abstract

We prove regularity bounds for Koszul cycles holding for every ideal of dimension ≤ 1 in a polynomial ring; see Theorem 3.5. In Theorem 4.7 we generalize the "c 1" lower bound for the Green-Lazarsfeld index of Veronese rings proved in (Bruns et al., arXiv:0902.2431) to the multihomogeneous setting. For the Koszul complex of the c-th power of the maximal ideal in a Koszul ring we prove that the cycles of homological degree t and internal degree t(c 1) belong to the t-th power of the module of 1-cycles; see Theorem 5.2. © Springer-Verlag Berlin Heidelberg 2011.