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 10.57.1. Let $S$ be a graded ring. A set of homogeneous elements $f_ i \in S_{+}$ generates $S$ as an algebra over $S_0$ if and only if they generate $S_{+}$ as an ideal of $S$.

Proof. If the $f_ i$ generate $S$ as an algebra over $S_0$ then every element in $S_{+}$ is a polynomial without constant term in the $f_ i$ and hence $S_{+}$ is generated by the $f_ i$ as an ideal. Conversely, suppose that $S_{+} = \sum Sf_ i$. We will prove that any element $f$ of $S$ can be written as a polynomial in the $f_ i$ with coefficients in $S_0$. It suffices to do this for homogeneous elements. Say $f$ has degree $d$. Then we may perform induction on $d$. The case $d = 0$ is immediate. If $d > 0$ then $f \in S_{+}$ hence we can write $f = \sum g_ i f_ i$ for some $g_ i \in S$. As $S$ is graded we can replace $g_ i$ by its homogeneous component of degree $d - \deg (f_ i)$. By induction we see that each $g_ i$ is a polynomial in the $f_ i$ and we win. $\square$


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