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Slight generalization of the main theorem of [Beauville-Laszlo].

Theorem 15.90.17. 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.

Proof. The functor is fully faithful due to the exactness of ( for glueable modules, which tells us exactly that $H^0 \circ \text{Can} = \text{id}$ on the full subcategory of glueable modules. Hence it suffices to check essential surjectivity. That is, we must show that an arbitrary glueing datum $(M', M_1, \alpha _1)$ arises from some glueable $R$-module.

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.

Put $M = H^0((M', M_1, \alpha _1)) = \mathop{\mathrm{Ker}}(\text{d})$. We obtain an exact sequence of $R$-modules
\begin{equation} \label{more-algebra-equation-define-M} 0 \to M \to M' \oplus M_1 \to (M')_ f \to 0. \end{equation}

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 gives rise to the original glueing datum.

Since $f$ is a nonzerodivisor on $M_1$, we have $M[f^\infty ] \cong M'[f^\infty ]$. This yields an exact sequence
\begin{equation} \label{more-algebra-equation-exact-mod-torsion} 0 \to M/M[f^\infty ] \to M_1 \to (M')_ f/M' \to 0. \end{equation}

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 15.90.16 we have $\text{Tor}_1^ R(R', \mathop{\mathrm{Coker}}(M' \to (M')_ f)) = 0$. The sequence ( thus remains exact upon tensoring over $R$ with $R'$. Using $\alpha _1$ and Lemma 15.90.2 the resulting exact sequence can be written as
\begin{equation} \label{more-algebra-equation-mod-torsion-sequence} 0 \to (M/M[f^\infty ]) \otimes _ R R' \to (M')_ f \to (M')_ f/M' \to 0 \end{equation}

This yields an isomorphism $(M/M[f^\infty ]) \otimes _ R R' \cong M'/M'[f^\infty ]$. This implies that in the diagram

\[ \xymatrix{ & M[f^\infty ] \otimes _ R R' \ar[r] \ar[d] & M \otimes _ R R' \ar[r] \ar[d] & (M/M[f^\infty ]) \otimes _ R R' \ar[r] \ar[d] & 0 \\ 0 \ar[r] & M'[f^\infty ] \ar[r] & M' \ar[r] & M'/M'[f^\infty ] \ar[r] & 0, } \]

the third vertical arrow is an isomorphism. Since the rows are exact and the first vertical arrow is an isomorphism by Lemma 15.90.2 and $M[f^\infty ] = M'[f^\infty ]$, the five lemma implies that $M \otimes _ R R' \to M'$ is an isomorphism. This completes the proof. $\square$

Comments (2)

Comment #5480 by Takagi Benseki(高城 辨積) on

In the second paragraph of the proof, should be ?

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