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arXiv · math/0701190

Betti numbers and shifts in minimal graded free resolutions

Tim Roemer
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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.

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Tim Roemer. 2007-01-06. Betti numbers and shifts in minimal graded free resolutions. https://doi.org/10.1215/ijm/1318598667

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