I was very confused, for proving the Hilbert-Serre thm. The following is our assumptions.
$A=\bigoplus_{n=0}^{\infty}A_n$ is graded Noetherian ring, so $A$ is finitely generated $A_0$-algebra, generated by homogeneous elements $x_1,\cdots,x_s$ each of which has degree $k_i$.
$M=\bigoplus_{n=1}^{\infty}M_n$ is finitely generated graded $A$-module, and each $M_n$ turns out to be a finitely generated $A_0$-module.
$\lambda$ is additive function on the class of finitely generated $A_0$-module.
$P$ is generating function w.r.t $\lambda(M_n)$, i.e., $P(M,t)=\Sigma\lambda(M_n)t^n \in \mathbb{Z}[[t]]$ ($\textit{Poincare series}$)
The proof is given by following.
Everything is clear for me, except for $\textit{how we define $L_n$ for small n}$?, especially $n$ is smaller than $k_s$. I strongly guess that we have to define $L_n=M_n$.
There is some more detailed proof of this theorem in google, but I can't still find what is exactly $L_n$ for small $n$. Can anyone tell me what is $L_n$ and if $L_n=M_n$ is true for such small $n$ as I mentioned, then can you tell me WHY L is a graded $A$-module? (Since $\textit{Poincare series}$ defined on graded $A$-module)