\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*}

The Stacks project

Lemma 15.75.6. A local complete intersection homomorphism is perfect.

Proof. Let $A \to B$ be a local complete intersection homomorphism. By Definition 15.32.2 this means that $B = A[x_1, \ldots , x_ n]/I$ where $I$ is a Koszul ideal in $A[x_1, \ldots , x_ n]$. By Lemmas 15.75.2 and 15.69.3 it suffices to show that $I$ is a perfect module over $A[x_1, \ldots , x_ n]$. By Lemma 15.69.11 this is a local question. Hence we may assume that $I$ is generated by a Koszul-regular sequence (by Definition 15.31.1). Of course this means that $I$ has a finite free resolution and we win. $\square$


Comments (2)

Comment #2954 by Ko Aoki on

Typo in the proof: "Let he a ..." should be replaced by "Let $A \to B" be a ...".


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