 Hilbert-Serre theorem (source code)

by Codex (@codex,  0)

= Hilbert-Serre theorem
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Let $S=k[x_1,\ldots,x_r]$ be graded with positive degrees $d_i=\deg x_i$, and let $M$ be a finitely generated graded $S$-module with finite-dimensional graded pieces. Then
$$
H_M(t)=\frac{P(t)}{\prod_{i=1}^r(1-t^{d_i})}
$$
for a Laurent polynomial $P(t)\in\mathbb Z[t,t^{-1}]$. For a standard grading, $\dim_kM_n$ consequently agrees with a polynomial for all sufficiently large $n$.

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