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 10.100.1. Let $(R, \mathfrak m)$ be a local ring with nilpotent maximal ideal $\mathfrak m$. Let $M$ be a flat $R$-module. If $A$ is a set and $x_\alpha \in M$, $\alpha \in A$ is a collection of elements of $M$, then the following are equivalent:

  1. $\{ \overline{x}_\alpha \} _{\alpha \in A}$ forms a basis for the vector space $M/\mathfrak mM$ over $R/\mathfrak m$, and

  2. $\{ x_\alpha \} _{\alpha \in A}$ forms a basis for $M$ over $R$.

Proof. The implication (2) $\Rightarrow $ (1) is immediate. Assume (1). By Nakayama's Lemma 10.19.1 the elements $x_\alpha $ generate $M$. Then one gets a short exact sequence

\[ 0 \to K \to \bigoplus \nolimits _{\alpha \in A} R \to M \to 0 \]

Tensoring with $R/\mathfrak m$ and using Lemma 10.38.12 we obtain $K/\mathfrak mK = 0$. By Nakayama's Lemma 10.19.1 we conclude $K = 0$. $\square$


Comments (2)

Comment #3621 by JuanPablo on

Alternative proof: generate by Nakayama's Lemma 00DV. Then one gets a short exact sequence . Then one tensors by and by Lemma 00HL one gets . Then by Nakayama's Lemma 00DV one gets .

There are also:

  • 2 comment(s) on Section 10.100: Flatness criteria over Artinian rings

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