and thm. 2.1.2).

\[vanishing\] Let $V(J) \subset Spec(P)$ the variety defined by the ideal $J$ and let $d:=dim(V(J))$. Then $H_k({\bf f})=0$ for $k >d$ and $H_d({\bf f}) \neq 0$

Let us first look at the case where $\dim(J)=0$. Then the above result tells us that $H_0({\bf f})=P/J$ and $H_k({\bf f})=0$ for $k>0$. So ${\bf f}$ is a regular sequence and the Koszul complex provides a free resolution of $P/J$ as a $P$-module. As the transpose of the map $ K_n {\longrightarrow}K_{n-1}$ can be identified, up to signs, with the map $K_1 {\longrightarrow}K_0$, we obtain an isomorphism $$Ext^n_P(P/J,P) \cong P/J$$

Recall Grothendiecks [*local duality theorem*]{} (see e.g. [@brunsherzog], thm. 3.5.8):

\[localduality\] Let $P$ be a $n$-dimensional local ring with maximal ideal ${{\mathfrak{m}}}$ and dualizing module $\omega_P$. For any finitely generated $P$-module $M$, there exists a natural non-degenerate pairing $$H^i_{{\mathfrak{m}}}(M) \times Ext^{n-i}_P(M,\omega_P) {\longrightarrow}H^n_{{\mathfrak{m}}}(\omega_P) \stackrel{tr}{{\longrightarrow}} K$$

In particular, for $M=P/J$ and $i=0$ we obtain a non-degenerate pairing $$P/J \times Ext^n_P(P/J,\omega_P) {\longrightarrow}K$$ as $H^0_{{\mathfrak{m}}}(P/J)=P/J$. The choice of an isomorphism $\phi: P \stackrel{\cong}{{\longrightarrow}} \omega_P$ induces isomorphisms $$Ext^n_P(P/J,\omega_P)\cong Ext^n_P(P/J,P)\cong P/J$$ then hence provides us with a perfect pairing $$B_{\phi}: P/J \times P/J {\longrightarrow}K$$ As $B_{\phi}$ is $P$-linear, it follows that it factors over the multiplication map 