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 10.129.8. Let $R$ be a ring. Let $R \to S$ be a ring map which is (a) flat, (b) of finite presentation, (c) has Cohen-Macaulay fibres. Then we can write $S = S_0 \times \ldots \times S_ n$ as a product of $R$-algebras $S_ d$ such that each $S_ d$ satisfies (a), (b), (c) and has all fibres equidimensional of dimension $d$.

Proof. For each integer $d$ denote $W_ d \subset \mathop{\mathrm{Spec}}(S)$ the set defined in Lemma 10.129.4. Clearly we have $\mathop{\mathrm{Spec}}(S) = \coprod W_ d$, and each $W_ d$ is open by the lemma we just quoted. Hence the result follows from Lemma 10.23.3. $\square$


Comments (2)

Comment #1562 by kollar on

I find the formulation confusing. I suggest:

Let R be a ring. Let R→Sd be ring maps. Assume that R→S0×…×Sn is (a) flat, (b) of finite presentation, (c) has Cohen-Macaulay fibres. Then each R→Sd satisfies (a), (b), (c) and has all fibres equidimensional of dimension d.

Comment #1580 by on

OK, I tried to clear up the statement. Thanks! The change is here.


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