New singularity invariants : the sheaf $beta_X^bullet$

The graded coherent sheaf $alpha_X^bullet$ constructed in [B.18] for any reduced pure dimensional complex space $X$ is stable by exterior product but not by the de Rham differential. We construct here a new graded coherent sheaf $alpha_X^bullet$ containing $alpha_X^bullet$ and stable both by exterior product and by the de Rham differential. We show that it has again the ``pull-back property for holomorphic maps $f : X to Y$ between irreducible complex spaces such that $f(X)$ is not contained in the singular set of $Y$. Moreover, this graded coherent sheaf $alpha_X^bullet$ comes with a natural coherent exhaustive filtration and this filtration is also compatible with the pull-back by such holomorphic maps. These sheaves define new invariants on singular complex spaces. We show on some simple examples that these invariants are new.