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*}

Remark 15.81.12. Let $(R \to R', f)$ be a glueing pair and let $M$ be an $R$-module. Here are some observations which can be used to determine whether $M$ is glueable for $(R \to R', f)$.

  1. By Lemma 15.81.11 we see that $M$ is glueable for $(R \to R^\wedge , f)$ if and only if $M[f^\infty ] \to M \otimes _ R R^\wedge $ is injective. This holds if $M[f] \to M^\wedge $ is injective, i.e., when $M[f] \cap \bigcap _{n = 1}^\infty f^ n M = 0$.

  2. If $\text{Tor}_1^ R(M, R'_ f) = 0$, then $M$ is glueable for $(R \to R', f)$ (use Algebra, Lemma 10.74.2). This is equivalent to saying that $\text{Tor}_1^ R(M, R')$ is $f$-power torsion. In particular, any flat $R$-module is glueable for $(R \to R', f)$.

  3. If $R \to R'$ is flat, then $\text{Tor}_1^ R(M, R') = 0$ for every $R$-module so every $R$-module is glueable for $(R \to R', f)$. This holds in particular when $R$ is Noetherian and $R' = R^\wedge $, see Algebra, Lemma 10.96.2


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