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Let $F$ be a filtered algebra and let $G$ be its associated graded algebra. Some examples of properties of $F$ that can be concluded from properties of $G$:

(A) The dimension of $F$ is equal to the dimension of $G$

(B) If $G$ is a Frobenius algebra then $F$ is a Frobenius algebra.

What are other properties of $F$ that can be concluded from properties of $G$?

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    $\begingroup$ Are you willing to assume that the filtration is exhaustive, separated/Hausdorff, and complete? Or perhaps just finite? If so, then the filtration on F induces a nice filtration on the bar resolution of F, and that means you get a nice spectral sequence in Ext, Tor, and various other homological invariants, going from the chosen invariant of G, to the chosen invariant of F. Differentials in the SS can wipe out classes, but not introduce new ones, so here's a property of F that can be concluded from a property of G: an upper bound on the size of a homological invariant, like $Ext^*$. $\endgroup$
    – user164898
    Commented Sep 5, 2022 at 16:53
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    $\begingroup$ The Noetherian property: see the answer here $\endgroup$
    – M T
    Commented Sep 6, 2022 at 9:10
  • $\begingroup$ If your filtration is finite dimensional, then the Gelfand-Kirillov dimension of $G$ gives the Gelfand-Kirillov dimension for $F$. $\endgroup$
    – jg1896
    Commented Apr 12, 2023 at 21:32

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Another interesting property is that if $G$ is free of nonzero zero-divisors, then so is $F$. For instance, this argument is used in the case of the universal enveloping algebra $U(L)$ of a Lie algebra $L$. As the associated graded algebra $G$ of $U(L)$ (with respect to the canonical filtration) is a polynomial ring, one concludes that $U(L)$ has no nonzero zero-divisors.

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