In the case $S=K[[t]]$ we obtain a commutative diagram of the following form $$\begin{array}{ccccccccc}
0&{\longrightarrow}&M_S          &\stackrel{t\cdot}{{\longrightarrow}}&M_S            &{\longrightarrow}& M_0          &{\longrightarrow}&0\\
 &    &\uparrow\cong&                        &\uparrow \cong&    &\uparrow \cong&    &\\
0&{\longrightarrow}&Ext^n_R(M_S,R)&\stackrel{t\cdot}{{\longrightarrow}}&Ext^n_R(M_S,R)      &{\longrightarrow}& Ext^n_P(M_0,P)&{\longrightarrow}&0\\
 &    &\uparrow\cong&                        &\uparrow \cong&    &\uparrow \cong&    &\\
0&{\longrightarrow}&Hom_S(M_S,S)  &\stackrel{t\cdot}{{\longrightarrow}}&Hom_S(M_S,S)   &{\longrightarrow}& Hom_K(M_0,K) &{\longrightarrow}&0\\
\end{array}$$

Application to non-isolated hypersurface singularity with one-dimensional critical locus
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Hypersurfaces with one-dimensional singular locus
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Following [@pellikaan1], the [*primitive*]{} of an ideal $I \subset P:={{\mathbb C}}\{x_1,\ldots,x_n\}$ is the ideal $$\int I:=\{f \in P\;|\;(f,\partial_1f,\ldots,\partial_nf)\subset I\}$$ This ideal arises when studying functions $f$ which contain a specific sub-space $\Sigma:=V(I)$ inside their critical locus. One can pursue the classification program of singularity theory in this context and Siersma [@siersma] started the investigation of [*line-singularities*]{}, which correspond to the case where $I$ is a radical ideal defining a line. The [*extended $I$-codimension*]{} of a function $f \in \int I$ is defined as $$c_{e,I}(f):=dim_{{{\mathbb C}}}(\int I/\int I \cap J_f)$$ and one can try to classify the cases of low codimension. Here $J_f=(\partial_1f,\ldots\partial_nf)$ is the jacobian ideal of $f$, [@pellikaan1]. As $J_f \subset I$, one can consider the associated [*jacobian module*]{} $I/J_f$.

We will assume from now on that $I$ is a radical ideal, defining a curve germ $\Sigma \subset ({{\mathbb C}}^n,0)$. On 