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*}

10.116 Dimension of graded algebras over a field

Here is a basic result.

Lemma 10.116.1. Let $k$ be a field. Let $S$ be a finitely generated graded algebra over $k$. Assume $S_0 = k$. Let $P(T) \in \mathbf{Q}[T]$ be the polynomial such that $\dim (S_ d) = P(d)$ for all $d \gg 0$. See Proposition 10.57.7. Then

  1. The irrelevant ideal $S_{+}$ is a maximal ideal $\mathfrak m$.

  2. Any minimal prime of $S$ is a homogeneous ideal and is contained in $S_{+} = \mathfrak m$.

  3. We have $\dim (S) = \deg (P) + 1 = \dim _ x\mathop{\mathrm{Spec}}(S)$ (with the convention that $\deg (0) = -1$) where $x$ is the point corresponding to the maximal ideal $S_{+} = \mathfrak m$.

  4. The Hilbert function of the local ring $R = S_{\mathfrak m}$ is equal to the Hilbert function of $S$.

Proof. The first statement is obvious. The second follows from Lemma 10.56.8. The equality $\dim (S) = \dim _ x\mathop{\mathrm{Spec}}(S)$ follows from the fact that every irreducible component passes through $x$ according to (2). Hence we may compute this dimension as the dimension of the local ring $R = S_{\mathfrak m}$ with $\mathfrak m = S_{+}$ by Lemma 10.113.6. Since $\mathfrak m^ d/\mathfrak m^{d + 1} \cong \mathfrak m^ dR/\mathfrak m^{d + 1}R$ we see that the Hilbert function of the local ring $R$ is equal to the Hilbert function of $S$, which is (4). We conclude the last equality of (3) by Proposition 10.59.8. $\square$


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