Remark 15.90.18. Let $(R \to R', f)$ be a glueing pair. Let $M$ be an $R$-module that is not necessarily glueable for $(R \to R', f)$. Setting $M' = M \otimes _ R R'$ and $M_1 = M_ f$ we obtain the glueing datum $\text{Can}(M) = (M', M_1, \text{can})$. Then $\tilde M = H^0(M', M_1, \text{can})$ is an $R$-module that is glueable for $(R \to R', f)$ and the canonical map $M \to \tilde M$ gives isomorphisms $M \otimes _ R R' \to \tilde M \otimes _ R R'$ and $M_ f \to \tilde M_ f$, see Theorem 15.90.17. From the exactness of the sequences

$M \to (M \otimes _ R R' )\oplus M_ f \to M \otimes _ R (R')_ f \to 0$

and

$0 \to \tilde M \to (\tilde M \otimes _ R R') \oplus \tilde M_ f \to \tilde M \otimes _ R (R')_ f \to 0$

we conclude that the map $M \to \tilde M$ is surjective.

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