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Let's suppose we have two noncommutative graded k-coalgebras $C_1$ and $C_2$ with respective admissible filtrations (i.e $F_{0}C_i=0$ and $\mathrm{colim}_k F_kC_i=C_i$), I would like to know if there is an isomorphism $$\mathrm{Gr}(C_1\prod C_2)\cong \mathrm{Gr}(C_1)\prod\mathrm{Gr}(C_2)$$ where the filtration in $C_1\prod C_2$ is $F_k(C_1\prod C_2)=\displaystyle\sum_{p+q=k}{F_pC_1\prod F_q C_2}$. I saw the dual result for k-algebras in page 190 of the book Cogroups and Co-rings in Categories of Associative Rings, but without proof. Any suggestion, please?.

By the way, $C_1\prod C_2=C_1\oplus C_2\oplus (C_1\otimes C_2)\oplus (C_2\otimes C_1)\oplus (C_1\otimes C_2\otimes C_1)\oplus\cdots$

the summands are alternating tensor products.

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    $\begingroup$ The title mentions coproducts and the notation $\prod$ stands for products. Is that products or coproducts of coalgebras you are asking about? $\endgroup$ Apr 18, 2019 at 18:30
  • $\begingroup$ Anyway, if you believe that whatever operation of two coalgebras you are interested in is computed by the formula you wrote after "by the way" -- isn't it obvious then that this commutes with the passage to the associated graded objects? Both the direct sums and the tensor products of vector spaces do. $\endgroup$ Apr 18, 2019 at 18:38
  • $\begingroup$ @leonid Sorry I actually meant product instead of coproduct in the title, already corrected. $\endgroup$
    – Victor TC
    Apr 19, 2019 at 12:56
  • $\begingroup$ @leonid This seems indeed evident if we define the filtration of $C_1\prod C_2$ as the induced by the expression in terms of direct sums and tensor product, but this filtration is not exactly defined in such way, unless eventualy they both agree but I am not aware of this last fact. $\endgroup$
    – Victor TC
    Apr 19, 2019 at 14:05
  • $\begingroup$ OK, now I see. Presuming that the formulas for the product and for the filtration on the product are correct, the question then seems to reduce essentially to this: let $V$ be a vector space with an increasing filtration $F$ (in the situation at hand, $V=C_1$ or $V=C_2$). Define a filtration on the tensor power $V^{\otimes n}$ (for some $n\ge1$) by the rule $F_p(V^{\otimes n})=(F_pV)^{\otimes n}$. How to compute the associated graded vector space to $V^{\otimes n}$ with this filtration $F$? Does the question reduce to this, indeed? $\endgroup$ Apr 19, 2019 at 18:05

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