I am reading this paper of Positselski and Vishik, in particular the main theorem: the cohomology algebra of a conilpotent algebra (i.e. the cohomology of its cobar construction) is Koszul if it satisfies certain conditions.

Consider a coaugmented coalgebra $C$ and filter $\overline C$ by its coradical filtration, denote it $\{F_p\overline C\}$. This produces a graded coalgebra $\operatorname{gr}(C)$ whose degree $i$ part is $F_iC/F_{i-1}C$. Moreover, the the cobar construction $\Omega C$ inherits a filtration so that

$$F_n\Omega^i C = \bigoplus_{\sum j_i = n} F_{j_1}\overline C\otimes \cdots \otimes F_{j_i}\overline C$$

This filtration is bounded below and exhaustive, so the spectral sequence associated to this filtration converges to $H^*(\Omega C)$ which, like in the paper, I will write just $H^*(C)$. If $C$ is graded, then write $H^{*,j}(C)$ for the space of cocycles of total weight $j$.

In the paper, the authors say that the $E_0$ page of this spectral sequence is the cobar construction of $\operatorname{gr}(C)$ and that $E_1^{i,j} = H^{i,j}(\operatorname{gr}(C))$. This doesn't fit the convention I'm used to, where the $E_0$-page has in bidegree $(p,q)$ the space $\Omega^{p+q}(\operatorname{gr}(C))$ in weight $p$ (by the very definition of the filtration given in the paper), so $E_1^{i,j}$ should instead be $H^{i+j,i}$. Thus, to obtain the claim of the authors, one would have to use the grading $(q,p-q)$.

There are still other issues: for example, I cannot see why $E_1^{1,0} = H^{1,1}(\operatorname{gr}(C))$ should coincide with $H^1(C) = E_\infty^{1,0}$. This should happen since $F_p H^1(C) = F_{p-1}H^1(C)$ for $p>1$. Indeed, $E_1^{p,1-p} = 0$ for such indices. This gives $E_\infty^{p,1-p} = 0$ for $p>1$ and $E_\infty^{1,0} = H^1(C)$, but it is not clear to me why $d_1 : E_1^{1,0}\to E_1^{2,0} = H^{2,2}(\operatorname{gr}(C))$ is zero.

Moreover, the paper claims that the differential $d_r$ has bidegree $(1,-r)$. But the differential of the spectral sequence of this filtration has differential $d_r$ of bidegree $(r,1-r)$. It is true, though, that with the regrading above at least $d_1$ now has bidegree $(1,1)$ (which is still not $(1,-1)$?), but still I cannot see why $d_1^{1,0}=0$.

Can someone clarify this? That is, what is the grading convention used for this spectral sequence? The filtration defined produces, according to what I know, a spectral sequence with a different grading convention. There are still a series of steps in the proof I cannot understand, but perhaps fixing the above grading issues helps with this.

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