---- > [!theorem] Theorem. ([[Poincare Series and Hilbert-Serre]]) > > Let $A=\bigoplus_{n \geq 0}A_{n}$ be a [[Noetherian ring|Noetherian]] [[graded ring]]. By [[characterizing Noetherianity in graded rings]], $A_{0}$ is a [[Noetherian ring]] and $A$ is [[subalgebra generated by a subset|finitely generated]] as an $A_{0}$-[[algebra]] by some $x_{1},\dots,x_{s}, x_{i} \in A_{k_{i}}, k_{i} >0$. > > Let $M=\bigoplus_{n \geq 0}M_{n}$ be a nonzero [[submodule generated by a subset|finitely generated]] [[graded module|graded]] $A$-[[module]]. Certainly each $M_{n}$ is an $A_{0}$-[[module]]; in fact, $M_{n}$ is a *finitely generated* $A_{0}$-[[module]]. Indeed: > - $M=\text{span}_{A}\{ m_{1},\dots,m_{t} \}, m_{i} \in M_{r_{i}}, r_{i} \geq 0$ > - Thus $M_{n}=\{ \sum_{i=1}^{t} a_{i}m_{i}: a_{i} \in A_{n-r_{i}} \}$ > - Thus $M_{n}=\text{span}_{A_{0}} \underbrace{ \left\{ x_{1}^{e_{1}} \cdots x_{s}^{e_{s}} \cdot m_{i}: 1 \leq s \leq t , \sum_{j=1}^{s}k _{j} e_{j} = n- r_{i} \right\} }_{ \text{finite} }.$ > From now on, we have another assumption: $\text{The ring }A_{0} \text{ is Artinian}.$ > Thus, each $M_{n}$ is a finitely generated module over an [[Artinian ring]] $A_{0}$. Since $M_{n}$ [[module is Noetherian (resp. Artinian) iff submodule and quotient is|is therefore]] both Noetherian and Artinian, $\ell(M_{n})<\infty$. > > > > [!definition] Definition. (Poincare Series) > > > > Let $A=\bigoplus_{n\geq 0}A$ be a [[Noetherian ring|Noetherian]] [[graded ring]], [[subalgebra generated by a subset|generated]] as an $A_{0}$-[[algebra]] by $x_{1},\dots,x_{s}, x_{i} \in A_{k_{i}}, k_{i}>0$, where $A_{0}$ is [[Artinian ring|Artinian]] (e.g., a [[field]]). Let $M=\bigoplus_{n \geq 0}M_{n}$ be a [[submodule generated by a subset|finitely generated]] [[graded module|graded]] $A$-[[module]]. The **Poincare series** of $M$ is the [[power series]] $P(M,T)=\sum_{n=0}^{\infty} \ell(M_{n})T^{n} \ \ \ \ \ \in \mathbb{Z}[ [T]].$ > > > [!specialization] Specialization. > > > If $A_{0}=k$ is a [[field]] then $M_{n}=V_{n}$ is a [[vector space]], $\ell(M_{n})=\text{dim}_{k}V_{n}$ and so $P(M,T)=P(V,T)$ takes the form $P(M,T)=\sum_{n=0}^{\infty} \text{dim}_{k}V_{n} \ T^{n} \ \ \in \mathbb{Z}[ [T]]. $ > > > > ^specialization > > The **Hilbert-Serre Theorem** states that the Poincare series of $M$ is a [[polynomial 4|polynomial]] with integer coefficients divided by the [[polynomial 4|polynomial]] $(1-T^{k_{1}})\cdots (1-T^{k_{s}})$. That is, $P(M,T)$ is a [[rational function]] of the form $\frac{f(T)}{\prod_{i=1}^{s}(1-T^{k_{i}})}, \ \ f(T) \in \mathbb{Z}[T].$ > > [!basicexample] The canonical example. > Let $k$ be a [[field]]. Consider the [[polynomial 4|polynomial ring]] $A=k[T_{1},\dots,T_{s}]=\bigoplus_{n \geq 0}A_{n}$, where $A_{n}$ is the additive subgroup consisting of $0$ and all [[homogeneous polynomial|homogeneous]] polynomials of degree $n$. > 1. $A$ is [[subalgebra generated by a subset|generated]] as an $A_{0}=k$-[[algebra]] by $T_{1},\dots,T_{s} \in A$. Thus $k_{1}=\dots k_{s}=1$. > 2. What is $\ell(A_{n})=\text{dim}_{k}A_{n}$? As usual, $A_{n}$ has a $k$-linear [[basis]] of degree-$n$ monomials in $k[T_{1},\dots,T_{s}]$. I count ${n+s-1\choose n}={n+s-1\choose s-1}$ of these. > 3. We have[^2] $\begin{align} > P(A,T)&= \sum_{n \geq 0} {n+s-1\choose s-1} T^{n} \\ > &= \frac{1}{(1-T)^{s}} > \end{align}$ > by the binomial theorem. This matches the expression which Hilbert-Serre predicted. > ^canonical-example [^2]: (Thinking of $A$ as a [[graded module]] over itself.) > [!proof]- Proof. ([[Poincare Series and Hilbert-Serre]]) > Summary: > - Induction of the number $s$ of generators for $A_{0}$. > - $s=0$: argue that, since $A=A_{0}$, $P(M,T)$ is itself a normal polynomial in this case > - Reindex $n \in \mathbb{Z}$ instead of $n \geq 0$ (just makes later stuff easier, put everything with negative index to zero) > - $s >0$: multiplication by $x_{s}$ gives the usual kernel-cokernel exact sequence in each degree. Argue that direct summing the degrees together gives two finitely-generated graded $A$-modules; in fact, f.g. graded modules over $A_{0}[x_{0},\dots,x_{s-1}]$. > - Apply $\ell$ to the sequence, getting the usual alternating sum thing for additive functors applied to exact sequences in each degree. Then sum over all $n \in \mathbb{Z}$. Some massaging and the induction hypothesis finish. > > Induction on the number $s$ of generators for $A_{0}$. > > **$s=0$.** In this case $P(M,T) \in \mathbb{Z}[T]$ (a polynomial). Indeed, in this case $A=A_{0}$. ![[Pasted image 20250511180148.png|100]] > So $M$ generated as an $A_{0}$-module by some finite subset $S \subset M$. Take $n_{0} \geq 0$ big enough that $S \subset M_{0} \oplus M_{1} \oplus \dots \oplus M_{n_{0}}$. Then $P(M,T)=\sum_{n=0}^{n_{0}} \ell(M_{n}) T^{n}$ > has finitely many terms, so is an earnest polynomial. > > > **$s>0$.** Assume the theorem holds for $s-1$ generators of $A_{0}$. Let $s \geq 2$. (Re)write $M=\bigoplus_{n \in \mathbb{Z}}M_{n}$, with $M_{\ell}=0$ for $\ell<0$. Let $n \in \mathbb{Z}$. Then $m \mapsto x_{s}m:M_{n} \to M_{n+k_{s}}$ is a [[linear map|homomorphism]] of $A_{0}$-[[module|modules]], giving rise thus to an [[exact sequence]] of $A_{0}$-[[module|modules]] $(*) \ 0 \to K_{n} \to M_{n} \xrightarrow{\cdot x_{s}}M_{n + {k_{s}}} \to L_{n+k_{s}} \to 0$ > where $K_{n}=\operatorname{ker}( m \mapsto x_{s}m)$ and $L_{n+k_{s}}=\operatorname{coker}(m \mapsto x_{s}m)$. Define > - $K=\bigoplus_{n \in \mathbb{Z}} K_{n}$ > - $L=\bigoplus_{n \in \mathbb{Z}}L_{n+k_{s}}$ > these are both [[graded module|graded]] $A$-[[module|modules]][^2] which are [[module is Noetherian (resp. Artinian) iff submodule and quotient is|Noetherian (finitely-generated is what we care about) because]] $M$ is. > > $K$ and $L$ are useful because $x_{s}$ [[annihilator of a module|annihilates]] them: by construction $x_{s}K=0=x_{s}L$.[^3] Consequently, we may regard $K,L$ as f.g. graded modules over $A_{0}[x_{1},\dots,x_{s-1}]$. Now recall $\ell$ is additive and ES1.12. Apply it to $(*)$ thus: $\ell(K_{n})-\ell(M_{n})+\ell(M_{n+k_{s}})-\ell(L_{n+k_{s}})=0.$ > Multiply this by $T^{n+k_{s}}$, and then write in the following special way: $\ell(M_{n+{k_{s}}})T^{n+k_{s}} - T^{k_{s}}\ell(M_{n})T^{n}=\ell(L_{n+k_{s}})T^{n+k_{s}}-T^{k_{s}}\ell(K_{n})T^{n}.$ > Then sum over all $n \in \mathbb{Z}$: $\underbrace{ P(M, T)-T^{k_{s}}P(M,T) }_{ (1-T^{k_{s}})P(M,T) }=\underbrace{ \underbrace{ P(L, T) }_{ \frac{f_{1}(T)}{\prod_{i=1}^{s-1}(1-T^{k_{i}})} }-\underbrace{ T^{k_{s}}P(K,T) }_{ \frac{f_{2}(T)}{\prod_{i=1}^{s-1} (1-T^{k_{i}})} } }_{ = \frac{f_{1}(T)-f_{2}(T)}{\prod_{i=1}^{s-1}(1-T^{k_{i}})} },$ > Where the reformulations of the RHS come from the induction hypothesis. Now divide by $(1-T^{k_{s}})$ to get the desired result. > > ---- #### [^3]: Check. [^2]: Check. ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```