Let S = K [ x 1, …, x n ] be a polynomial ring and R = S / I where I ? S is a graded ideal. The Multiplicity Conjecture of Herzog, Huneke, and Srinivasan which was recently proved using the Boij–S?derberg 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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机译：令S = K [x 1 ，…，x n ]为多项式环，并且R = S / I其中I？ S是理想的等级。最近使用Boij–S？derberg理论证明的Herzog，Huneke和Srinivasan的多重猜想指出，R的多重性也受R相对于S的最小最小自由分辨率的最大位移的函数所限制如果R为Cohen–Macaulay，则由最小位移的函数所限定。在本文中，我们研究了相关的问题，以表明R的总贝蒂数也由R的最小渐变自由分辨率中的偏移函数所限制，并且在R的情况下也由偏移的另一函数所限制是科恩·马考雷。我们还将讨论这些界限很明显的情况。
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