Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring and $R=S/I$ where $I \subset S$ is a graded ideal. The Multiplicity Conjecture of Herzog, Huneke, and Srinivasan which was recently proved using the Boij-Söderberg 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.
"Betti numbers and shifts in minimal graded free resolutions." Illinois J. Math. 54 (2) 449 - 467, Summer 2010. https://doi.org/10.1215/ijm/1318598667