The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.57.10. Let $k$ be a field. Suppose that $I \subset k[X_1, \ldots , X_ d]$ is a nonzero graded ideal. Let $M = k[X_1, \ldots , X_ d]/I$. Then the numerical polynomial $n \mapsto \dim _ k(M_ n)$ (see Example 10.57.9) has degree $ < d - 1$ (or is zero if $d = 1$).

Proof. The numerical polynomial associated to the graded module $k[X_1, \ldots , X_ d]$ is $n \mapsto \binom {n - 1 + d}{d - 1}$. For any nonzero homogeneous $f \in I$ of degree $e$ and any degree $n >> e$ we have $I_ n \supset f \cdot k[X_1, \ldots , X_ d]_{n-e}$ and hence $\dim _ k(I_ n) \geq \binom {n - e - 1 + d}{d - 1}$. Hence $\dim _ k(M_ n) \leq \binom {n - 1 + d}{d - 1} - \binom {n - e - 1 + d}{d - 1}$. We win because the last expression has degree $ < d - 1$ (or is zero if $d = 1$). $\square$


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