# Which information can be obtained from Poincaré series ?

If $A= \bigoplus_{i\ge 0}A_i$ is a graded commutative Noetherian algebra over a field, its Poincaré series is given by $P(t) = \sum_{i\ge 0} \dim(A_i)t^i$. Although the definition of $P(t)$ only depends on the graded vector space underlying $A$, the Krull dimension of the ring $A$ can be obtained from the Poincaré series as the order of the pole of $P(t)$ at $t=1$.

Question: Are there other information about the ring structure of $A$ that can be obtained from $P(t)$ ?

Since I'm particularly interested in cases where $A$ is the cohomology ring of a finite group, I'm also looking for an example of finite groups whose mod-p cohomology rings are not isomorphic but have identical Poincaré series.

I don't know if it's possible to obtain further information on the ring structure in general. However, if $A$ is the mod-p cohomology ring of a finite group, a result of Benson and Carlson states that if $A$ is Cohen-Macaulay, then $P(t)$ satisfies the functional equation $$P(1/t) = (-1)^d P(t)\hspace{90pt}(\ast)$$ where $d$ is the Krull dimension of $A$. Conversely, if a given $P(t)$ doesn't satisfy this equation, you know that $A$ isn't Cohen-Macaulay, i.e. the depth of $A$ is less than its Krull dimension.

In case $A$ has Krull dimension 2, than $A$ is Cohen-Macaulay iff $P(t)$ satisfies $(\ast)$. These results can be found in the paper Benson, Carlson: Functional equations for Poincaré series in group cohomology. Bull. London Math. Society 26(1994), 438-448.

As an example for groups with identical Poincaré series you can take $E := \mathbb{Z}/2 \times \mathbb{Z}/2$ and $D_8$, the dihedral group of order 8. Their cohomology rings $$H^\ast(E;\mathbb{F}_2)=\mathbb{F}_2[x,y],\;\; |x|=|y|=1$$ $$H^\ast(D_8;\mathbb{F}_2)=\mathbb{F}_2[x,y,z]/(xy),\;\; |x|=|y|=1, |z|=2$$ aren't isomorphic (since only one is a domain) but both have Poincaré series $P(t)=\frac{1}{(1-t)^2}$.

Added: There is a paper of R. Stanley (who is also active on MO) that contains some properties of the Poincaré series that may be of interest. I just quote a few:

• If $A$ is Gorenstein of Krull dimension $d$, then $P(1/t)=(-1)^dt^aP(t)$ for some integer $a$. This generalizes $(\ast)$ (with a=0) because the mod-p cohomology ring of a finite group is Cohen-Macaulay iff its Gorenstein.

• $A$ is a complete intersection with generators of degree 1 iff $P(t)$ has the form $$P(t)=\frac{\prod_{i=1}^l(1+t+\cdots+t^{m_i})}{(1-t)^d}$$

• A sequence of homogeneous elements $x_1,...,x_k \in A$ of positive degrees $n_i$ is regular iff $P(A,t) = P(B,t)/\prod_i (1-t^{n_i})$ where $B=A/(x_1,...,x_n)$.

• In general, I expect there to be statements of the form "nice rings have nice Poincar\'e series", so in the direction of your title, "nasty Poincar\'e series implies nasty ring". Jul 6, 2012 at 17:05
• @Allen: What do you mean by "nice rings" resp. "nice Poincaré series" ? Can you give me some more details on the statements you expect to be there. References are also welcome. Thanks.
– tj_
Jul 6, 2012 at 21:57