Spectral sequences of a Morse shelling

We recently introduced a notion of tilings of the geometric realization of a finite simplicial complex and related those tilings to the discrete Morse theory of R. Forman, especially when they have the property to be shellable, a property shared by the classical shellable complexes. We now observe that every such tiling supports a quiver which is acyclic precisely when the tiling is shellable and then that every shelling induces two spectral sequences which converge to the (co)homology of the complex. Their first pages are free modules over the critical tiles of the tiling.