Let S be a polynomial ring and R=S/I where I is a graded ideal of S. TheMultiplicity Conjecture of Herzog, Huneke, and Srinivasan which was recentlyproved using the Boij-Soederberg theory states that the multiplicity of R isbounded above by a function of the maximal shifts in the minimal graded freeresolution of R over S as well as bounded below by a function of the minimalshifts if R is Cohen-Macaulay. In this paper we study the related problem toshow that the total Betti-numbers of R are also bounded above by a function ofthe shifts in the minimal graded free resolution of R as well as bounded belowby another function of the shifts if R is Cohen-Macaulay. We also discuss thecases when these bounds are sharp.
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机译：令S为一个多项式环，R = S / I，其中I为S的理想点。最近使用Boij-Soederberg理论证明的Herzog，Huneke和Srinivasan的多重猜想指出，R的多重性在一个函数的上方R在S上的最小渐变自由分辨率中的最大偏移的最大偏移，如果R是Cohen-Macaulay，则在下面以最小偏移的函数为边界。在本文中，我们研究了相关的问题，以表明R的总贝蒂数也受R的最小渐变自由分辨率的偏移的函数限制在上方，以及如果R为Cohen-Macaulay，则受偏移的另一函数限制。我们还讨论了这些界限很明确的情况。
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