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Let $R=\mathbb{Q}[x_1,x_2,\ldots,x_n]$ and $I$ a non-homogeneous ideal of $R$. Then the algebra R/I is filtered. It has an associated graded algebra gr(R/I).

Let $I_1$ be the initial ideal of $I$, see for example the paper. Then $I_1$ is a homogeneous ideal of $R$ and $R/I_1$ is a graded algebra.

Is the Hilbert series of gr(R/I) the same as the Hilbert series of $R/I_1$? Thank you very much.

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    $\begingroup$ The answer is affirmative: a general fact that isn't hard to prove is that $\mathrm{gr}(R/I)$ and $R/\mathrm{in}(I)$ are isomorphic as (graded) algebras. But (as per question mathoverflow.net/questions/325312/…) I'm lacking a good reference =) $\endgroup$
    – Igor Makhlin
    Commented Jul 30, 2019 at 1:38

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