Free resolutions of parameter ideals for some rings with finite local cohomology

Abstract:

Let $R$ be a $d$-dimensional local ring, with maximal ideal $\mathfrak {m}$, containing a field and let $x_1, \dots , x_d$ be a system of parameters for $R$. If $\operatorname {depth} R \geq d - 1$ and the local cohomology module $\operatorname {H}_{m}^{d-1}(R)$ is finitely generated, then there exists an integer $n$ such that the modules $R/(x_1^i,\dots ,x_d^i)$ have the same Betti numbers, for all $i\geq n$.