The Stacks project

Lemma 10.58.1. 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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