Abstract
Let R = k [ x 1 , … , x n ] be a polynomial ring and let I ⊂ R be a graded ideal. In [T. Römer, Betti numbers and shifts in minimal graded free resolutions, arXiv: AC/070119], Römer asked whether under the Cohen–Macaulay assumption the ith Betti number β i ( R / I ) can be bounded above by a function of the maximal shifts in the minimal graded free R-resolution of R / I as well as bounded below by a function of the minimal shifts. The goal of this paper is to establish such bounds for graded Cohen–Macaulay algebras k [ x 1 , … , x n ] / I when I is a standard determinantal ideal of arbitrary codimension. We also discuss other examples as well as when these bounds are sharp.
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