Koszul Homology Under Small Perturbations

Let $x_1,\ldots,x_s$ be a filter regular sequence in a local ring $(R,\mathfrak{m})$. Denote by $R_{x_1,\ldots,x_s}$ the Koszul complex of $x_1,\ldots,x_s$ over $R$. In this paper, we give an explicit number $N$ such that the sum of lengths $\sum_{i=1}^s (-1)^i\ell(H_i(R_{x_1,\ldots,x_s}))$ is preserved when we perturb the sequence $x_1, \ldots,x_s$ by $\varepsilon_1, \ldots, \varepsilon_s \in \mathfrak{m}^N$. Applying this result and the main Theorem of Eisenbud, we show that there exits $N >0$ such that for all $i \geq 1$ the length of $H_i(R_{x_1,\ldots,x_s})$ is preserved under small perturbation.