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 10.134.13. Let $R$ be a ring. Let $S = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ be a relative global complete intersection. For every prime $\mathfrak q$ of $S$, let $\mathfrak q'$ denote the corresponding prime of $R[x_1, \ldots , x_ n]$. Then

  1. $f_1, \ldots , f_ c$ is a regular sequence in the local ring $R[x_1, \ldots , x_ n]_{\mathfrak q'}$,

  2. each of the rings $R[x_1, \ldots , x_ n]_{\mathfrak q'}/(f_1, \ldots , f_ i)$ is flat over $R$, and

  3. the $S$-module $(f_1, \ldots , f_ c)/(f_1, \ldots , f_ c)^2$ is free with basis given by the elements $f_ i \bmod (f_1, \ldots , f_ c)^2$.

Proof. First, by Lemma 10.68.2, part (3) follows from part (1). Parts (1) and (2) immediately reduce to the Noetherian case by Lemma 10.134.12 (some minor details omitted). Assume $R$ is Noetherian. Let $\mathfrak p = R \cap \mathfrak q'$. By Lemma 10.133.4 for example we see that $f_1, \ldots , f_ c$ form a regular sequence in the local ring $R[x_1, \ldots , x_ n]_{\mathfrak q'} \otimes _ R \kappa (\mathfrak p)$. Moreover, the local ring $R[x_1, \ldots , x_ n]_{\mathfrak q'}$ is flat over $R_{\mathfrak p}$. Since $R$, and hence $R[x_1, \ldots , x_ n]_{\mathfrak q'}$ is Noetherian we may apply Lemma 10.98.3 to conclude. $\square$


Comments (2)

Comment #3254 by Dario WeiƟmann on

We could define . It is clear from the context, but still.

There are also:

  • 2 comment(s) on Section 10.134: Syntomic morphisms

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