Betti numbers of determinantal ideals

Details Betti numbers of determinantal ideals Betti numbers of determinantal ideals

2007-01-16

arxiv.org/abs/math/0701435v1

Mathematics

Commutative Algebra

Scientific paper

Let $R=k[x_1, ..., x_n]$ be a polynomial ring and let $I\subset R$ be a graded ideal. In \cite{R}, R\"{o}mer asked whether under the Cohen-Macaulay assumption the $i$-th Betti number $\beta_{i}(R/I)$ 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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