residue pairings*]{}, introduced by K. Saito, which express the self-duality of the Gau[ß]{}-Manin system of the singularity, [@saito].

Homology and Cohomology
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We have seen that in the case of a regular sequence ${\bf f}$, the origin of the pairing on $P/J$ lies in the self-dual nature of the Koszul complex. We now investigate in general what consequence this self-duality has for the Koszul homology groups.

\[selfdual\] (c.f. [@eisenbud], prop. 17.15) Let ${\bf f}=f_1,f_2,\ldots,f_n$ be a sequence of elements in a ring $P$. There are isomorphisms $$\alpha_p: H_p\left({\bf f}\right)\cong H^{n-p}\left({\bf f}\right), \;\;p=0,1,\ldots,n,$$ where $H^p({\bf f}):= H^p(Hom_P\left(K_{\bullet}\left({\bf f}\right),P\right))$ is the Koszul cohomology.

Using the basis $e_1,e_2,\ldots,e_n$ for the free module $P^n$ and the dual basis $\phi_1,\ldots,\phi_n$ for the dual module $Hom_P(P^n,P)$, one obtains basis elements $e_I:=e_{i_1}\wedge e_{i_2} \wedge \ldots \wedge e_{i_p}$ for $\bigwedge^p P^n$ and $\phi_I=\phi_{i_1}\wedge\ldots \wedge\phi_{i_p}$, $(I=(i_1,i_2,\ldots,i_p))$ for the dual module $Hom_P(\bigwedge^p P^n,P)$. Define isomorphisms $\alpha_p:\bigwedge P^p {\longrightarrow}(\bigwedge^pP^n)^*:=Hom_P(\bigwedge^{n-p}P^n,P)$ by setting $\alpha_p(e_I)=sign(\sigma)\phi_J$ where $J$ is the sequence of indices complementary to $I$ and $sign(\sigma)$ is the sign of the permutation that puts the sequence $(I,J)$ into $(1,2,\ldots,n)$. One verifies that in this way one obtains a mapping between complexes $\bigwedge^{\bullet} P^n$ and $(\bigwedge^{\bullet} P^n)^*$.

We refer to [@eisenbud], section 17.4 for 