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\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 10.133.12. Let

\[ \xymatrix{ B & S \ar[l] \\ A \ar[u] & R \ar[l] \ar[u] } \]

be a commutative square of local rings. Assume

  1. $R$ and $\overline{S} = S/\mathfrak m_ R S$ are regular local rings,

  2. $A = R/I$ and $B = S/J$ for some ideals $I$, $J$,

  3. $J \subset S$ and $\overline{J} = J/\mathfrak m_ R \cap J \subset \overline{S}$ are generated by regular sequences, and

  4. $A \to B$ and $R \to S$ are flat.

Then $I$ is generated by a regular sequence.

Proof. Set $\overline{B} = B/\mathfrak m_ RB = B/\mathfrak m_ AB$ so that $\overline{B} = \overline{S}/\overline{J}$. Let $f_1, \ldots , f_{\overline{c}} \in J$ be elements such that $\overline{f}_1, \ldots , \overline{f}_{\overline{c}} \in \overline{J}$ form a regular sequence generating $\overline{J}$. Note that $\overline{c} = \dim (\overline{S}) - \dim (\overline{B})$, see Lemma 10.133.6. By Lemma 10.98.3 the ring $S/(f_1, \ldots , f_{\overline{c}})$ is flat over $R$. Hence $S/(f_1, \ldots , f_{\overline{c}}) + IS$ is flat over $A$. The map $S/(f_1, \ldots , f_{\overline{c}}) + IS \to B$ is therefore a surjection of finite $S/IS$-modules flat over $A$ which is an isomorphism modulo $\mathfrak m_ A$, and hence an isomorphism by Lemma 10.98.1. In other words, $J = (f_1, \ldots , f_{\overline{c}}) + IS$.

By Lemma 10.133.6 again the ideal $J$ is generated by a regular sequence of $c = \dim (S) - \dim (B)$ elements. Hence $J/\mathfrak m_ SJ$ is a vector space of dimension $c$. By the description of $J$ above there exist $g_1, \ldots , g_{c - \overline{c}} \in I$ such that $J$ is generated by $f_1, \ldots , f_{\overline{c}}, g_1, \ldots , g_{c - \overline{c}}$ (use Nakayama's Lemma 10.19.1). Consider the ring $A' = R/(g_1, \ldots , g_{c - \overline{c}})$ and the surjection $A' \to A$. We see from the above that $B = S/(f_1, \ldots , f_{\overline{c}}, g_1, \ldots , g_{c - \overline{c}})$ is flat over $A'$ (as $S/(f_1, \ldots , f_{\overline{c}})$ is flat over $R$). Hence $A' \to B$ is injective (as it is faithfully flat, see Lemma 10.38.17). Since this map factors through $A$ we get $A' = A$. Note that $\dim (B) = \dim (A) + \dim (\overline{B})$, and $\dim (S) = \dim (R) + \dim (\overline{S})$, see Lemma 10.111.7. Hence $c - \overline{c} = \dim (R) -\dim (A)$ by elementary algebra. Thus $I = (g_1, \ldots , g_{c - \overline{c}})$ is generated by a regular sequence according to Lemma 10.133.6. $\square$


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