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 10.134.17. Let $R \to S$, $S \to S'$ be ring maps.

  1. If $R \to S$ and $S \to S'$ are syntomic, then $R \to S'$ is syntomic.

  2. If $R \to S$ and $S \to S'$ are relative global complete intersections, then $R \to S'$ is a relative global complete intersection.

Proof. Assume $R \to S$ and $S \to S'$ are syntomic. This implies that $R \to S'$ is flat by Lemma 10.38.4. It also implies that $R \to S'$ is of finite presentation by Lemma 10.6.2. Thus it suffices to show that the fibres of $R \to S'$ are local complete intersections. Choose a prime $\mathfrak p \subset R$. We have a factorization

\[ \kappa (\mathfrak p) \to S \otimes _ R \kappa (\mathfrak p) \to S' \otimes _ R \kappa (\mathfrak p). \]

By assumption $S \otimes _ R \kappa (\mathfrak p)$ is a local complete intersection, and by Lemma 10.134.3 we see that $S \otimes _ R \kappa (\mathfrak p)$ is syntomic over $S \otimes _ R \kappa (\mathfrak p)$. After replacing $S$ by $S \otimes _ R \kappa (\mathfrak p)$ and $S'$ by $S' \otimes _ R \kappa (\mathfrak p)$ we may assume that $R$ is a field. Say $R = k$.

Choose a prime $\mathfrak q' \subset S'$ lying over the prime $\mathfrak q$ of $S$. Our goal is to find a $g' \in S'$, $g' \not\in \mathfrak q'$ such that $S'_{g'}$ is a global complete intersection over $k$. Choose a $g \in S$, $g \not\in \mathfrak q$ such that $S_ g = k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ is a global complete intersection over $k$. Since $S_ g \to S'_ g$ is still syntomic also, and $g \not\in \mathfrak q'$ we may replace $S$ by $S_ g$ and $S'$ by $S'_ g$ and assume that $S = k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ is a global complete intersection over $k$. Next we choose a $g' \in S'$, $g' \not\in \mathfrak q'$ such that $S' = S[y_1, \ldots , y_ m]/(h_1, \ldots , h_ d)$ is a relative global complete intersection over $S$. Hence we have reduced to part (2) of the lemma.

Suppose that $R \to S$ and $S \to S'$ are relative global complete intersections. Say $S = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ and $S' = S[y_1, \ldots , y_ m]/(h_1, \ldots , h_ d)$. Then

\[ S' \cong R[x_1, \ldots , x_ n, y_1, \ldots , y_ m]/(f_1, \ldots , f_ c, h'_1, \ldots , h'_ d) \]

for some lifts $h_ j' \in R[x_1, \ldots , x_ n, y_1, \ldots , y_ m]$ of the $h_ j$. Hence it suffices to bound the dimensions of the fibres. Thus we may yet again assume $R = k$ is a field. In this case we see that we have a ring, namely $S$, which is of finite type over $k$ and equidimensional of dimension $n - c$, and a finite type ring map $S \to S'$ all of whose nonempty fibre rings are equidimensional of dimension $m - d$. Then, by Lemma 10.111.6 for example applied to localizations at maximal ideals of $S'$, we see that $\dim (S') \leq n - c + m - d$ as desired. $\square$


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