intersections
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We now make the further assumption, that $dim(P)=n$, so that the almost complete intersection ${\bf f}:=f_1,\ldots,f_n$ gives an ideal $J=(f_1,f_2,\ldots,f_n)$ that defines a one-dimensional locus $V(J) \subset Spec(P)$.

\[koszhom\] Let ${\bf f}:=f_1,\ldots,f_n$ define a one-dimensional almost complete intersection. Then:\
(i) $$Ext^{n-1}_P(H_0({\bf f}),P)\cong H_1({\bf f}),\;\;Ext_P^{n}(H_1({\bf f}),P)=0.$$ (ii) There is an exact sequence $$0 {\longrightarrow}Ext^{n}_P(H_0,P) {\longrightarrow}H_0({\bf f}){\longrightarrow}Ext^{n-1}_P(H_1({\bf f}),P) {\longrightarrow}0$$

As $P$ is a assumed to be a regular local ring of dimension $n$, we have $Ext^k_P(-,P)=0$ for $k > n$. So the result follows from the above proposition \[aci\]

\[imodj\] For a one-dimensional almost complete intersection defined by ${\bf f}$ we put $$M:=Ext^{n}_P(H_0({\bf f}),P)$$ As $Ext^{n}_P(H_0({\bf f}),P) \subset H_0({\bf f})=P/J$ we see that $$M=I/J$$ for an ideal $I \subset P$.

In other words, $I$ is defined by having an isomorphism of exact sequences $$\begin{array}{cccccccccc}
0 &{\longrightarrow}&Ext^{n}_P(H_0({\bf f}),P)& {\longrightarrow}&H_0({\bf f})&{\longrightarrow}& Ext^{n-1}_P(H_1({\bf f}),P)& {\longrightarrow}& 0\\
&&\cong\downarrow&&\cong \downarrow&&\cong\downarrow&&\\
0&{\longrightarrow}&I/J&{\longrightarrow}&P/J&{\longrightarrow}&P/I&{\longrightarrow}& 0\\
\end{array}$$

\[saturation\] The ideal $I$ is equal to the [*saturation*]{} of $J$ with respect to the maximal ideal: $$I=\bigcup_k(J:\mathfrak{m}^k) \supset J$$ Thus the module $M=I/J$ is the *${\mathfrak{m}}$-torsion submodule of $P/J$*: $$M=I/J=H^0_{{\mathfrak{m}}}(P/J)$$

Because $Ext^{n}_P(H_1({\bf f}),P)=0$, we see that the module $H_1({\bf f})$ has $Ext^{n-1}_P(H_1({\bf f}),P)=P/I$ as the only non-vanishing $Ext$. Hence it is 