Hilbert–Poincaré series

Hilbert–Serre theorem

Suppose M is a finitely generated graded module over ${\displaystyle A[x_1,\dots ,x_n],\mathrm{deg}{x}_i=d_i}$ with an Artinian ring (e.g., a field) A. Then the Poincaré series of M is a polynomial with integral coefficients divided by ${\displaystyle \prod (1-t^{d_i})}$. The standard proof today is an induction on n. Hilbert's original proof made a use of Hilbert's syzygy theorem (a projective resolution of M), which gives more homological information.

Here is a proof by induction on the number n of indeterminates. If ${\displaystyle n=0}$, then, since M has finite length, ${\displaystyle M_k=0}$ if k is large enough. Next, suppose the theorem is true for ${\displaystyle n-1}$ and consider the exact sequence of graded modules (exact degree-wise), with the notation ${\displaystyle N(l{)}_k=N_{k+l}}$,

${\displaystyle 0\to K(-d_n)\to M(-d_n)\stackrel{x_n}{\to }M\to C\to 0}$.

Since the length is additive, Poincaré series are also additive. Hence, we have:

${\displaystyle P(M,t)=-P(K(-d_n),t)+P(M(-d_n),t)-P(C,t)}$.

We can write ${\displaystyle P(M(-d_n),t)=t^{d_n}P(M,t)}$. Since K is killed by ${\displaystyle x_n}$, we can regard it as a graded module over ${\displaystyle A[x_0,\dots ,x_{n-1}]}$; the same is true for C. The theorem thus now follows from the inductive hypothesis.

Chain complex

An example of graded vector space is associated to a chain complex, or cochain complex C of vector spaces; the latter takes the form

${\displaystyle 0\to C^0\stackrel{d_0}{⟶}{C}^1\stackrel{d_1}{⟶}{C}^2\stackrel{d_2}{⟶}\cdots \stackrel{d_{n-1}}{⟶}{C}^n⟶0.}$

The Hilbert–Poincaré series (here often called the Poincaré polynomial) of the graded vector space ${\displaystyle \underset{i}{⨁}{C}^i}$ for this complex is

${\displaystyle P_C(t)=\sum _{j=0}^n\dim (C^j)t^j.}$

The Hilbert–Poincaré polynomial of the cohomology, with cohomology spaces H^j = H^j(C), is

${\displaystyle P_H(t)=\sum _{j=0}^n\dim (H^j)t^j.}$

A famous relation between the two is that there is a polynomial ${\displaystyle Q(t)}$ with non-negative coefficients, such that ${\displaystyle P_C(t)-P_H(t)=(1+t)Q(t).}$