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 49.3.4. Let $A$ be a ring. Let $f \in A$. Let $X$ be a scheme over $\mathop{\mathrm{Spec}}(A)$. Let

\[ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 \]

be an inverse system of $\mathcal{O}_ X$-modules. Assume

  1. either $H^1(X, \mathcal{F}_1)$ is an $A$-module of finite length or $A$ is Noetherian and $H^1(X, \mathcal{F}_1)$ is a finite $A$-module,

  2. the equivalent conditions of Lemma 49.3.1 hold.

Then the inverse system $M_ n = \Gamma (X, \mathcal{F}_ n)$ satisfies the Mittag-Leffler condition.

Proof. Set $I = (f)$. We will use the criterion of Lemma 49.2.1. Observe that $f^ n : \mathcal{F}_0 \to I^ n\mathcal{F}_{n + 1}$ is an isomorphism for all $n \geq 0$. Thus it suffices to show that

\[ \bigoplus \nolimits _{n \geq 1} H^1(X, \mathcal{F}_1) \cdot f^{n + 1} \]

is a graded $S = \bigoplus _{n \geq 0} A/(f) \cdot f^ n$-module satisfying the ascending chain condition. If $A$ is not Noetherian, then $H^1(X, \mathcal{F}_1)$ has finite length and the result holds. If $A$ is Noetherian, then $S$ is a Noetherian ring and the result holds as the module is finite over $S$ by the assumed finiteness of $H^1(X, \mathcal{F}_1)$. Some details omitted. $\square$


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