Proposition 8

Suppose the sequence x is regular for M. The element of $\mathrm{Ext}_A^n(M/(x)M, M)$ associated with the exact sequence (23) is $(-1)^{n(n+1)/2} \theta_x$. For $i = 0, 1, \ldots, n$, define an A-linear mapping

as follows: if $m \in \mathbf{K}^i(x, M)$, $a^i(m)$ is the class in $M_i$ of the element $m(1, 2, \ldots, i)$ of $M$. It is clear that $a^0$ is the identity mapping of $M$ and that $p \circ a^n = q$. Moreover $a^{i+1} \circ \partial^i(m)$ is the image in $M_{i+1}$ of the element Since $x_k$ annihilates $M_{i+1}$ for $k = 1, \ldots, i$, $a^{i+1} \circ \partial^i(m)$ is also the image of hence By X, p. 120, cor. 1 and 2, the element of $\mathrm{Ext}_A^n(M/(x)M, M)$ associated with (23) is equal to $\prod_{i=1}^n (-1)^i \cdot \theta_x$, whence the assertion.