BETTI NUMBERS AND SHIFTS IN MINIMAL GRADED FREE RESOLUTIONS

Abstract

Let S = K[x(1), ... , x(n)] be a polynomial ring and R=S/1 where 1 subset of S is a graded ideal. The Multiplicity Conjecture of Herzog. Huneke, and Srinivasan winch was recently proved using the Boij-Soderberg 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 its 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.