The Stacks project

Lemma 15.25.3. Let $R$ be a ring. Let $S = R[x_1, \ldots , x_ n]$ be a graded polynomial algebra over $R$, i.e., $\deg (x_ i) > 0$ but not necessarily equal to $1$. Let $M$ be a graded $S$-module. Assume

  1. $R$ is a local ring,

  2. $M$ is a finite $S$-module, and

  3. $M$ is flat over $R$.

Then $M$ is finitely presented as an $S$-module.

Proof. Let $M = \bigoplus M_ d$ be the grading on $M$. Pick homogeneous generators $m_1, \ldots , m_ r \in M$ of $M$. Say $\deg (m_ i) = d_ i \in \mathbf{Z}$. This gives us a presentation

\[ 0 \to K \to \bigoplus \nolimits _{i = 1, \ldots , r} S(-d_ i) \to M \to 0 \]

which in each degree $d$ leads to the short exact sequence

\[ 0 \to K_ d \to \bigoplus \nolimits _{i = 1, \ldots , r} S_{d - d_ i} \to M_ d \to 0. \]

By assumption each $M_ d$ is a finite flat $R$-module. By Algebra, Lemma 10.78.5 this implies each $M_ d$ is a finite free $R$-module. Hence we see each $K_ d$ is a finite $R$-module. Also each $K_ d$ is flat over $R$ by Algebra, Lemma 10.39.13. Hence we conclude that each $K_ d$ is finite free by Algebra, Lemma 10.78.5 again.

Let $\mathfrak m$ be the maximal ideal of $R$. By the flatness of $M$ over $R$ the short exact sequences above remain short exact after tensoring with $\kappa = \kappa (\mathfrak m)$. As the ring $S \otimes _ R \kappa $ is Noetherian we see that there exist homogeneous elements $k_1, \ldots , k_ t \in K$ such that the images $\overline{k}_ j$ generate $K \otimes _ R \kappa $ over $S \otimes _ R \kappa $. Say $\deg (k_ j) = e_ j$. Thus for any $d$ the map

\[ \bigoplus \nolimits _{j = 1, \ldots , t} S_{d - e_ j} \longrightarrow K_ d \]

becomes surjective after tensoring with $\kappa $. By Nakayama's lemma (Algebra, Lemma 10.20.1) this implies the map is surjective over $R$. Hence $K$ is generated by $k_1, \ldots , k_ t$ over $S$ and we win. $\square$

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