The Coherence of direct images of relative De Rham complex

We show the coherence of direct images of relative De Rham complex for holomorphic map $\Phi$ with a suitable boundary condition. When the map $\Phi$ is proper, the result is classical and is well-known since the entries of Hodge to De Rham spectral sequence is already coherent. Therefore, our interest is the case for a non-proper (open) map which possesses critical points (not necessarily isolated). Then, the result implies certain finitely generatedness of vanishing cycles for the critical points. For the proof, we develop a new concept of Koszul-De Rham algebra, which seems to be of interest by itself.