Theorem 15.90.16. Let (R \to R',f) be a glueing pair. The functor \text{Can} : \text{Mod}_ R \longrightarrow \text{Glue}(R \to R', f) determines an equivalence of the category of R-modules glueable for (R \to R', f) and the category \text{Glue}(R \to R', f) of glueing data.
Slight generalization of the main theorem of [Beauville-Laszlo].
Proof. Let (M', M_1, \alpha _1) be a glueing datum. We will show that M = H^0((M', M_1, \alpha _1)) is a glueable for (R \to R', f) and that (M', M_1, \alpha _1) \cong \text{Can}(M).
We first check that the map \text{d} : M' \oplus M_1 \to (M')_ f used in the definition of the functor H^0 is surjective. Observe that (x, y) \in M' \oplus M_1 maps to \text{d}(x, y) = x/1 - \alpha _1^{-1}(y \otimes 1) in (M')_ f. If z \in (M')_ f, then we can write \alpha _1(z) = \sum y_ i \otimes g_ i with g_ i \in R' and y_ i \in M_1. Write \alpha _1^{-1}(y_ i \otimes 1) = y_ i'/f^ n for some y'_ i \in M' and n \geq 0 (we can pick the same n for all i). Write g_ i = a_ i + f^ n b_ i with a_ i \in R and b_ i \in R'. Then with y = \sum a_ i y_ i \in M_1 and x = \sum b_ i y'_ i \in M' we have \text{d}(x, -y) = z as desired.
Since M = H^0((M', M_1, \alpha _1)) = \mathop{\mathrm{Ker}}(\text{d}) we obtain an exact sequence of R-modules
We will prove that the maps M \to M' and M \to M_1 induce isomorphisms M \otimes _ R R' \to M' and M \otimes _ R R_ f \to M_1. This will imply that M is glueable for (R \to R', f) and \text{Can}(M) \cong (M', M_1, \alpha _1) as desired.
Since f is a nonzerodivisor on M_1, we have M[f^\infty ] \cong M'[f^\infty ]. This yields an exact sequence
Since R \to R_ f is flat, we may tensor this exact sequence with R_ f to deduce that M \otimes _ R R_ f = (M/M[f^\infty ]) \otimes _ R R_ f \to M_1 is an isomorphism.
By Lemma 15.90.15 we have \text{Tor}_1^ R(R', \mathop{\mathrm{Coker}}(M' \to (M')_ f)) = 0. The sequence (15.90.16.2) thus remains exact upon tensoring over R with R'. Using \alpha _1 and Lemma 15.88.8 the resulting exact sequence can be written as
This yields an isomorphism (M/M[f^\infty ]) \otimes _ R R' \cong M'/M'[f^\infty ]. This implies that in the diagram
the third vertical arrow is an isomorphism. Since the rows are exact and the first vertical arrow is an isomorphism by Lemma 15.88.8 and M[f^\infty ] = M'[f^\infty ], the five lemma implies that M \otimes _ R R' \to M' is an isomorphism.
The above shows that \text{Can} is essentially surjective and that the functor H^0 maps into the category of glueable modules. Due to the exactness of (15.90.9.1) for glueable modules we have H^0 \circ \text{Can} = \text{id} on the category of glueable modules. This implies \text{Can} is fully faithful by Lemma 15.89.11 combined with Categories, Lemma 4.24.4. This finishes the proof. \square
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