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 10.108.12. Let $R$ be a ring. The following are equivalent

  1. $R$ has finite global dimension $\leq n$,

  2. every finite $R$-module has projective dimension $\leq n$, and

  3. every cyclic $R$-module $R/I$ has projective dimension $\leq n$.

Proof. It is clear that (1) $\Rightarrow $ (2) and (2) $\Rightarrow $ (3). Assume (3). Choose a set $E \subset M$ of generators of $M$. Choose a well ordering on $E$. For $e \in E$ denote $M_ e$ the submodule of $M$ generated by the elements $e' \in E$ with $e' \leq e$. Then $M = \bigcup _{e \in E} M_ e$. Note that for each $e \in E$ the quotient

\[ M_ e/\bigcup \nolimits _{e' < e} M_{e'} \]

is either zero or generated by one element, hence has projective dimension $\leq n$ by (3). By Lemma 10.108.11 this means that $M$ has projective dimension $\leq n$. $\square$


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