If C is a cocomplete coalgebra, then $\psi:C\rightarrow B\Omega C$ is a filtered quasi-isomorphism

Question

I am reading the PhD thesis thesis of Kenji Lefèvre-Hasegawa and the corresponding errata by Bernhard Keller, my question is about the first error found in the thesis. Lemma 1.3.2.3 c states 'the adjunction morphism $\psi:C\rightarrow B\Omega C$ is a filtered quasi-isomorphism' (C is endowed with the primitive filtration). Along the way of the proof it is claimed that the comultiplication on $\mathrm{Gr}C$ vanishes, and this is wrong. If we denote $W=\mathrm{Gr}(S^{-1}C)$, the thesis claims

$\mathrm{Gr}(B\Omega C)\xrightarrow{\backsim} B\Omega(\mathrm{Gr} C)$ is the sum of complexes $V_{i}=\displaystyle\bigoplus_{i_{1}+\cdots+i_{k}=i}{SW^{\otimes i_{1}}}\otimes\dots {SW^{\otimes i_{k}}}$

Are $\mathrm{Gr}(B\Omega C)$ and $B\Omega(\mathrm{Gr} C)$ indeed isomorphic?. If so, can anybody please give some details of such isomorphism?. I think it is consequence of the erroneous assupmtion on $\mathrm{Gr} C$, but I am not sure.

On the other hand, Keller's correction of the original proof uses a decreasing filtration on $B\Omega C$ named $F^{'}_{l}$, by describing

$F^{'}_{l}$ is generated by all $m$th tensor powers of $C$ where $m\geq l$

I find the description of $F^{'}_{l}$ clear enough but I do not know how decreasing filtrations work (may they induce isomorphism in homology?). I would really appreciate some clues about these inquiries and a short clarification of what is either rescued or discarded from the original proof, in the corrected proof.

Answer 1

If $\mathrm{Gr}(B\Omega C)\xrightarrow{\backsim} B\Omega(\mathrm{Gr} C)$ actually means they both are not isomorphic but there is a canonical quasi-isomorphism from $\mathrm{Gr}(B\Omega C)$ to $B\Omega(\mathrm{Gr} C)$, I may conclude a similar statement from an argument of spectral sequences, see A primer on spectral sequences, example 4.3 (a homomorphism $B\Omega(\mathrm{Gr}C)\rightarrow \mathrm{Gr}(B\Omega C)$ that induces an isomorphism in homology). I have no idea about an homomorphism in the opposite direction.