a Cohen-Macaulay $P$-module of dimension one. It follows that $H^0_{{\mathfrak{m}}}(P/I)=0$ and thus the artinian module $M=Ext^{n}_P(H_0({\bf f}),P)$ is equal to $H^0_{{\mathfrak{m}}}(P/J)$. Hence $I$ is nothing but the saturation of $J$ with respect to ${{\mathfrak{m}}}$.

In fact this provides an easy was to compute $I$ and $I/J$ using a computer algebra system like [Singular]{} or [Macaulay]{}. The ring $P/I$ obtained by dividing out the [**m**]{}-primary torsion of $P/J$ is sometimes called the [*Cohen-Macaulayfication*]{} of $P/J$.

Using $I/J$ and $P/I$ we can give a slight reinterpretation of the $Ext$ of Koszul homology.

\[extiso\] The map $P/J {\longrightarrow}P/I$ induces an isomorphism $$Ext^{n-1}_P(P/I,P) \cong Ext^{n-1}_P(P/J,P)$$ and the map $I/J {\longrightarrow}P/J$ induces an isomorphism $$Ext^{n}_P(P/J,P) \cong Ext^n_P(I/J,P)$$

Apply $Hom_P(\bullet,P)$ to the short exact sequence of $P$-modules $0{\longrightarrow}I/J{\longrightarrow}P/J{\longrightarrow}P/I{\longrightarrow}0$. As $I/J$ is ${\mathfrak{m}}$-torsion and $P/I$ is ${{\mathfrak{m}}}$-torsion free the result follows.

We now have all the ingredients for the following central result.

\[isoextij\] Let the sequence ${\bf f}=f_1,\ldots,f_n$ define a one-dimensional almost complete intersection $J$, and $M=I/J=H^0_{\bf m}(P/J)$. Then there is an isomorphism $$Ext^n_P(M,P) \cong M.$$

Combining theorem \[extiso\] and theorem \[koszhom\](i) we obtain an isomorphism $$Ext^{n-1}_P(P/I,P)\cong Ext^{n-1}_P(P/J,P) \cong H_1({\bf f}),$$ showing that $H_1({\bf f})$ is isomorphic to the dualising module of $P/I$. Because $P/I$ is Cohen-Macaulay, 