Koszul Homology Under Small Perturbations

Let $x_1,\dots ,x_s$ be a filter regular sequence in a local ring $(R,\mathfrak{m})$. Denote by $R_{x_1,\dots ,x_s}$ the Koszul complex of $x_1,\dots ,x_s$ over $R$. In this paper, we give an explicit number $N$ such that the sum of lengths ${\sum }_{i=1}^s(-1{)}^i\mathrm{\ell }(H_i(R_{x_1,\dots ,x_s}))$ is preserved when we perturb the sequence $x_1,\dots ,x_s$ by ${\epsilon }_1,\dots ,{\epsilon }_s\in {\mathfrak{m}}^N$. Applying this result and the main Theorem of Eisenbud, we show that there exits $N>0$ such that for all $i\ge 1$ the length of $H_i(R_{x_1,\dots ,x_s})$ is preserved under small perturbation.