# Is the Hilbert series of an ideal related to the Hilbert series of its homogenization?

Suppose we have a field $$k$$ of characteristic 0, let $$I$$ be an ideal of $$R=k[x_1,...,x_n]$$, and let $$H$$ be the homogenization of $$I$$ in $$S=R[z]$$. Is there any relationship between the Hilbert series of $$R/I$$ and the Hilbert series of $$S/H$$?

I know that the Hilbert series for $$R/I$$ can be computed via Gröbner bases, but I'm hoping to gain information about a Gröbner basis for $$I$$ from the Hilbert series of $$R/I$$. In particular, if $$R/I$$ has Krull dimension 0, then I would like to compute $$\dim_k(R/I)$$ without relying on Gröbner bases. The hope is that if the Hilbert series for $$R/I$$ and $$S/H$$ are related, then I can use some of the techniques for homogeneous ideals to work in the non-homogeneous case.

• Sorry for asking a stupid question, but if $I$ is not homogeneous then what is the Hilbert series of $R/I$? – Zach Teitler Jan 18 at 1:34
• Good question! The first answer is that I had just been using Macaulay2 to compute this, so I didn’t know off the top of my head. The second answer is that $R/I$ is filtered by degree, and the Hilbert series is the Hilbert series of the associated graded $\mathrm{gr}(R/I)$. – Stephen McKean Jan 18 at 5:11
• When I was double checking this definition, I saw a post that mentioned that the Hilbert series of $R/I$ is the same as the Hilbert series of $R/\mathrm{in}(I)$, but it feels like I would again need a Gröbner basis to exploit this. – Stephen McKean Jan 18 at 5:14