Spectral sequences associated to deformations

Lapin has constructed a multiplicative spectral sequence from a deformation of an -algebra. In particular, as noted by the same author, one can apply this construction to a deformation induced by a filtration of an -algebra. A question that naturally appears is whether this latter multiplicative spectral sequence is isomorphic to the one that is canonically associated to the filtration and that typically appears in basic textbooks on homological algebra. We provide a positive answer to the previous question and we also explain the interesting constructions of Lapin in more structural terms.