and is of the form $B_{\phi}(a,b)=\beta(a\cdot b)$ for some linear form $$\beta: P/J {\longrightarrow}K$$ The linear space $ker(\beta)$ is the [*socle*]{} of $P/J$, that is, its unique minimal ideal. There is a classical result of Scheja and Storch that states that this socle has a [ canonical generator]{}.

[@schejastorch] If $char(K)=0$ and $x_1,\ldots,x_n$ are generators of the maximal ideal, then the Jacobian determinant

$$h:=\left|\begin{array}{ccc} \frac{\partial f_1}{\partial x_1}& \ldots &\frac{\partial f_1}{\partial x_n}\\
\vdots&&\vdots\\
\frac{\partial f_n}{\partial x_1} &\ldots& \frac{\partial f_n}{\partial x_n}\\
 \end{array}\right|$$ is a generator for the socle of $P/J$.

In the case where $f \in P$ and $f_i=\partial f/\partial x_i$, the complex can be identified (up to some signs) with the $df \wedge$-complex $$0 {\longrightarrow}P \stackrel{df \wedge }{{\longrightarrow}}\Omega^1 \stackrel{df\wedge}{{\longrightarrow}} \ldots {\longrightarrow}\Omega^{n-1}
\stackrel{df \wedge}{{\longrightarrow}}\Omega^n {\longrightarrow}0$$ and the natural pairing takes the form $$B: \Omega^n/df\wedge \Omega^{n-1} \times\;\; \Omega^n/df\wedge \Omega^{n-1} {\longrightarrow}K$$ It is usually called [*residue pairing*]{} and has an analytic expression as $$B(P\omega,Q\omega)=\frac{1}{(2\pi i)^n}\int_{T_{\epsilon}} \frac{PQ\omega}{\partial_1f\ldots \partial_n f}$$ where $\omega:=dx_1\wedge dx_2 \wedge \ldots \wedge dx_n$ and $T_{\epsilon}:=\{x\;|\;|\partial_i f|=\epsilon\}$, $0<\epsilon \ll 1$. In the papers [@giventalvarchenko], [@varchenko] the relation between this pairing and the Poincaré-pairing in the cohomology of the Milnor fibre is described. The pairing is also the first in a sequence of [*higher 