Local Koszul H One Detects First Regularity Failure

Let ${\mathbf{x}}^{\mathrm{\prime }}=(x_1,\dots ,x_{j-1})$ and $N=M\mathrm{/}({\mathbf{x}}^{\mathrm{\prime }})M$. The preceding prefix is regular, so $K({\mathbf{x}}^{\mathrm{\prime }};M)$ has zero positive homology and $H_0=N$. The failure at $x_j$ gives $0\mathrm{\ne }{\mathrm{Ann}}_N(x_j)$. The cone exact sequence therefore identifies $H_1(K({\mathbf{x}}^{\mathrm{\prime }},x_j;M))$ with this nonzero annihilator.

Append the remaining entries one at a time. If $L$ is the current nonzero first homology and the next entry is $y\in \mathfrak{m}$, the cone exact sequence injects $L\mathrm{/}yL$ into the new first homology. The module $L$ is finite because it is homology of a bounded complex of finite modules over a Noetherian ring, and Nakayama gives $L\mathrm{/}yL\mathrm{\ne }0$. Thus first homology remains nonzero through every later entry, proving $H_1(K(\mathbf{x};M))\mathrm{\ne }0$.