I'm trying to understand a part of the PhD thesis of Kenji Lefèvre-Hasegawa (e.g. available [here](http://arxiv.org/abs/math/0310337)). My question is about the proof of Lemma 1.3.2.3b stating:



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> Remarquons que nous avons un isomorphisme de complexes $\bigoplus\_{i\geq 1}Gr\_i(\Omega BA)\to \Omega \overline{T^c}V$ qui identifie à la composante $Gr\_i(\Omega BA), i\geq 1$, à la somme des $S^{-1} V^{\otimes i\_1}\otimes\dots\otimes S^{-1} V^{\otimes i\_k}\subset (S^{-1} \overline{T^c}V)^{\otimes k}$, où $k\geq 1$ et où $i\_1+\dots+i\_k=i$.
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Here $V=SA$, where $S$ is the shift functor. $A$ is a dg algebra and $B$ and $\Omega$ are the bar and cobar resolution, respectively, i.e. $BA=\bigoplus\_{i\geq 1} V^{\otimes i}$ (with the coproduct $\Delta: V^{\otimes i}\to \bigoplus\_{p+q=i} V^{\otimes p}\otimes V^{\otimes q}$ and the filtration $BA\_{[i]}=ker\Delta^{(i)}$) and $\Omega BA=\bigoplus\_{i\geq 1}S^{-1}BA^{\otimes i}$ with the filtration induced by that of $BA$. 


Can someone give more details?



Associated graded of a filtration of a tensor product
OK let me try a naive answer, and then maybe you will elaborate a bit on what it is that worries you? 


The key idea is very simple - the bar differential has a part coming from the differential on $A$, and the remaining part, encoding the product of $A$. The first part preserves the number of tensor factors, the remaining part consists of terms with fewer tensor factors. Therefore, for the filtration by the number of tensor factors, the corresponding graded object will just remember the differential on $A$, that is the structure of a chain complex, not the product. It remains to notice that for obvious reasons the number of tensor factors is precisely what is counted by the filtration by kernels of iterated deconcatenations... Does it help?


