  Let $\Bbbk$ be a field of characteristic 0. Let $X$ be a smooth complete
intersection over $k$ of dimension $n-k$ in the projective space
$\mathbf{P}^n_{k}$, for given positive integers $n$ and $k$. When
$k=\mathbb{C}$, Terasoma (\cite{Ter90}) and Konno (\cite{Ko91}) provided an
explicit representative (in terms of differential forms) of a basis for the
primitive middle-dimensional algebraic de Rham cohomology
$H_{dR,\operatorname{prim}}^{n-k}(X;\mathbb{C})$. Later Dimca constructed
another explicit representative of a basis of
$H_{dR,\operatorname{prim}}^{n-k}(X;\mathbb{C})$ in \cite{Dim95}. Moreover, he
proved that his representative gives the same cohomology class as the previous
representative of Terasoma and Konno. The goal of this article is to examine
the above two different approaches without assuming that $k=\mathbb{C}$ and
provide a similar comparison result for any field $k$. Dimca's argument depends
heavily on the condition $k=\mathbb{C}$ and our idea is to find appropriate
Cech-de Rham complexes and spectral sequences corresponding to those two
approaches, which work without restrictions on $k$.
