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. 


Thanks in advance.



Which information can be obtained from 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.](http://homepages.abdn.ac.uk/mth192/pages/html/archive/benson-carlson.html)


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 
$P(t)=\frac{1}{(1-t)^2}$. 




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**Added:** There is a [paper of R. Stanley](http://math.mit.edu/~rstan/pubs/pubfiles/30.pdf) (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 $B=A/(x\_1,...,x\_n)$.


