Characterizing finite length local cohomology in terms of bounds on Koszul cohomology

Let $(R,m, \kappa)$ be a local ring. We give a characterization of $R$-modules $M$ whose local cohomology is finite length up to some index in terms of asymptotic vanishing of Koszul cohomology on parameter ideals up to the same index. In particular, we show that a quasi-unmixed module $M$ is asymptotically Cohen-Macaulay if and only if $M$ is Cohen-Macaulay on the punctured spectrum if and only if $\sup\{\ell(H^i(f_1, \ldots, f_d;M))\mid \sqrt{f_1, \ldots, f_d} = m \mbox{, } i< d\}<\infty$.