Resolutions corresponding to closed mappings

We consider certain resolutions defined for any closed mapping (they determine certain new spectral sequences). We first prove the naturality of the resolutions in their arguments. Next, we show that when the coefficient domain $L$ is a ring, the term $D^0$ is a sheaf of rings, and all the $D^p$ for $p\geqslant1$ are $D^0$-modules. This allows us to determine typical conditions for the resolution to be soft. It is shown that the resolution $D^*$ is a simplicial object in the sense of Eilenberg–Zilber, and a direct definition is given for multiplication in $D^*$. For completeness, an outline is given for the proof of the following fact (communicated to the author by A. V. Zarelua): in the case of a regular finite-sheeted covering, the spectral sequence corresponding to the resolution $D^*$ is isomorphic to the Cartan spectral sequence.
Bibliography: 8 titles.