plays a role in the Dwork-Griffiths description of the Hodge-pieces of the cohomology. For example, let $f \in {{\mathbb C}}[x_0,x_1,x_2,x_3,x_4]$ be the equation of a projective threefold $X=V(f) \subset {{\mathbb P}}^4$ of degree $d$ with only nodes as singularities, and let $\pi:Y {\longrightarrow}X$ a small resolution. Then the degree $a=2d-5$ part $(I/J)_a$ of $I/J$ can be identified with $H^{2,1}:=H^1(\Omega^2_Y)$ via $$A {\longrightarrow}\pi^*(Res(\frac{A\Omega}{f}))$$ where $\Omega:=\iota_E(dx_0 \wedge \ldots \wedge dx_4)$. Similarly, the degree $b=3d-5$ part of $I/J$ is identified with $H^{1,2}:=H^{2}(\Omega^1_Y)$, making a commutative diagram $$\begin{array}{ccccc}
(I/J)_a & \times & (I/J)_b & {\longrightarrow}&{{\mathbb C}}\\
\downarrow &&\downarrow&&\cong \downarrow\\
H^{2,1}&\times&H^{1,2}&\stackrel{\cup}{{\longrightarrow}}&H^{3,3}\\
\end{array}$$ For details we refer to [@dimcasaitowotzlaw].

Behaviour under flat deformations
=================================

We now study the behaviour of the module $I/J$ under deformation. By this we mean that we let the almost complete interesection $f_1,\ldots,f_n$ depend on additional parameters. If we require $H_0=P/(f_1,f_2,\ldots,f_n)$ to deform in a flat way, then the same will be true for $I/J$.\

Consider a local ring $S$ with maximal ideal ${\mathfrak{m}}_S$ and $K=S/{\mathfrak{m}}_S$ and let $R$ be a flat $S$-algebra such that $P=K \otimes_S R$. A sequence ${\bf F}:=F_1,F_2,\ldots,F_n \in R$ is called relative almost complete intersection if\
1) $$f_i:=F_i\;\; \textup{mod}\;\;{\mathfrak{m}}_S$$ 2) $H_0({\bf F})$ is $S$-flat.

\[flat\] If ${\bf 