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I was looking at this question and asked my self the following:

Let $A$ be graded algebra, which is also an $\mathbb{N}_0$-filtered algebra. If its associated graded algebra $\mathrm{gr}(A)$ is Koszul does that imply that $A$ is also Koszul?

Is this true for some suitable assumptions on the compatibility of the grading and the filtration?

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    $\begingroup$ What definition of Koszul do you want to use for such $A$? The usual definition requires that the algebra be graded... $\endgroup$
    – Manny Reyes
    Commented Apr 6, 2023 at 16:45
  • $\begingroup$ Yes $A$ should be graded. I have now added this to the question. $\endgroup$
    – Didier de Montblazon
    Commented Apr 6, 2023 at 17:49
  • $\begingroup$ Is there any relation between the grading and the filtration of $A$? Is it the canonical filtration associated to the grading? $\endgroup$
    – M.G.
    Commented Apr 6, 2023 at 17:56
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    $\begingroup$ @M.G. I am interested in the case where it is a different filtration. $\endgroup$
    – Didier de Montblazon
    Commented Apr 6, 2023 at 18:15

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