The Stacks project

Remark 15.90.12. 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 15.90.11 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 10.75.2). 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 10.97.2

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