Betti numbers and shifts in minimal graded free resolutions
arXiv:math/0701190 · doi:10.1215/ijm/1318598667
Abstract
Let S be a polynomial ring and R=S/I where I is a graded ideal of S. The Multiplicity Conjecture of Herzog, Huneke, and Srinivasan which was recently proved using the Boij-Soederberg theory states that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S as well as bounded below by a function of the minimal shifts if R is Cohen-Macaulay. In this paper we study the related problem to show that the total Betti-numbers of R are also bounded above by a function of the shifts in the minimal graded free resolution of R as well as bounded below by another function of the shifts if R is Cohen-Macaulay. We also discuss the cases when these bounds are sharp.
16 pages; corrected version