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I am preparing for an exam of commutative algebra, and I am at loss about how to compute Hilbert-Poincaré series of rings. In particular, I have some preparation exercises I can't solve. Mainly they involve computing the Poincaré series of quotient rings $A/I$. Two examples are:

Find the Hilbert-Poincaré series of the ring $\mathbb{C}[x,y,z]/(x^3+y^3+z^3)$.

and

Find the Hilbert-Poincaré series of the ring $\mathbb{C}[x,y,z,w]/I$ where $I=(x,y)\cap(z,w)$.

Any hint about how to proceed?

Daniel Robert-Nicoud
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1 Answers1

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In the first case set $R=\mathbb C[x,y,z]$ and $f=x^3+y^3+z^3$ is a homogeneous polynomial of degree 3. Consider the graded exact sequence $0\to R(-3)\stackrel{\cdot f}\to R\to R/fR\to 0$ and this shows that $H_{R/fR}(t)=(1-t^3)H_R(t)$.

In the second case $I$ is an intersection of monomial ideals, so $I$ is also a monomial ideal. In fact, $I=(xz,xw,yz,yw)$. Now it easy to compute $H_{R/I}(t)$: count the surviving monomials of degree $d$ in $R/I$ and observe that these are $x^iy^j$, respectively $z^iw^j$ with $i+j=d$. Thus we get $H_{R/I}(t)=1+\sum_{d\ge 1} (2d+2)t^d$. (I leave to you the finding of Hilbert series as a rational fraction.)

user26857
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  • Looks good. To answer the more general question of how to find Hilbert Series algorithmically, in full generality this is an active area of research in computational commutative algebra, but you could start by taking a look in to Grobner bases. Basically for these cases though, the slightly more general way to say what user121097 is doing is finding a graded free resolution ( ie. syzygies ) and using the fact that the Hilbert Series of a module ( eg. quotient ring ) is the alternating sum of the Hilbert series of the resolution. If you can find the resolution by inspection, all the easier. – Callus - Reinstate Monica Jan 23 '14 at 23:38
  • @Callus Let me see if I have understood correctly. For my second example, the resolution would be $0\to R(-3)^4\to R(-2)^4\to R\to R/I\to 0$, and the resulting series $H_{R/I}(t)=(-1+4t^2-4t^3)H_R(t)$. Correct? – Daniel Robert-Nicoud Jan 24 '14 at 00:40
  • @Callus Correction of my comment above: I forgot an element of the sequence on the left (a $R(-4)$) and got a sign wrong. The series should be $(1-4t^2+4t^3-t^4)H_R(t)$. – Daniel Robert-Nicoud Jan 24 '14 at 09:56
  • I still invite you to show your data to the mods. I really don't care. (Btw, only someone having serious problems collects data about other users. Or maybe a whistleblowing wannabe. In both cases, a pathetic human being.) – user26857 Feb 23 '24 at 19:45