Let $k$ be a field and $(C,\mathcal{O}\_C)$ be a smooth geometrically irreducible projective curve over $k$ of function field $k(C)$ and let $x$ be a closed point on it. From Laszlo-Beauville's lemma, the functor
$$\mathcal{F}\longmapsto (\mathcal{F}(C\setminus\{x\}),\mathcal{F}\_x,\iota\_x)$$
from the category of coherent sheaves of $\mathcal{O}\_C$-modules to the category of triplets $(M,N,\iota)$ where $M$ is a finitely generated $\mathcal{O}\_C(C\setminus\{x\})$, $N$ is a finitely generated $\mathcal{O}\_{C,x}$-module and $\iota\_x:M\otimes\_{\mathcal{O}\_C(C\setminus\{x\})}k(C)\to N\otimes\_{\mathcal{O}\_{C,x}}k(C)$ is an isomorphism of vector spaces over $k(C)$ (which corresponds to the gluing data in the latter case) is an equivalence of categories.


Let $R$ be a Dedekind domain which is also a $k$-algebra. I want to know whether the same machinery applies for $C\_R:=C\times\_k \operatorname{Spec} R$ with the subscheme $\{x\}\times\_k \operatorname{Spec}R$. Namely, given a triplet $(M,N,\iota)$ where $M$ is a finitely generated $\mathcal{O}\_C(C\setminus\{x\})\otimes R$-module, $N$ is a finitely generated $\mathcal{O}\_{C,x}\otimes R$ and $\iota$ is a $Q:=\operatorname{Quot}(k(C)\otimes\_k R)$-linear isomorphism from $M\otimes\_{\mathcal{O}\_{C}(C\setminus \{x\})\otimes\_k R} Q$ to $N\otimes\_{\mathcal{O}\_{C,x}\otimes\_k R} Q$, can I get a corresponding coherent sheaf on $C\_R$?


Sorry for the heavy notations. Any "elementary" answer is very welcome!


Many thanks!



Glueing modules over $\{x\}\times \operatorname{Spec} R$
The Beauville-Laszlo theorem holds in much greater generality - see Tag [0BNI](https://stacks.math.columbia.edu/tag/0BNI "0BNI") on the Stacks Project.


Let $A$ be any ring and let $f\in A$ be a non-zero divisor. Then the category of $f$-torsion free $A$-modules $M$ is equivalent to the category of triples $(M\_1, M\_2, \varphi)$ where $M\_1$ is an $A\_f$-module, $M\_2$ is an $f$-torsion free module over the $f$-adic completion $\widehat{A}$ of $A$, and $\phi \colon M\_1 \otimes\_{A\_f} (\widehat{A})\_f \xrightarrow{\sim} M\_2 \otimes\_{\widehat{A}} (\widehat{A})\_f$ is an isomorphism.


The equivalence sends an $A$-module $M$ to $(M\_f, \varprojlim\_n M/f^n M, \phi)$, where $\phi$ is the natural isomorphism. 


Furthermore, this equivalence respects the full subcategories of finitely generated, flat, and finite projective modules on either side. 


This has the following consequence: let $X$ be a smooth relative curve with geometrically connected fibers over *any* scheme $S$. Take any section $x \in X(S)$ and let $\Gamma\_x \hookrightarrow X$ be the image of $x$. Define $U$ to be the open subset $X - \Gamma\_x$ and $\widehat{X}$ as the formal completion of $X$ along $\Gamma\_x$. (In particular, we could take a smooth geometrically connected curve $X\_0$ over some field $k$ and let $X = X\_0 \times\_k S$, $x \in X\_0(S) = X(S)$.)


Then the natural restriction maps give an equivalence of categories from the category of finitely generated quasi coherent sheaves on $X\_S$ without sections vanishing along $\Gamma\_x$ (resp. flat f.g. quasicoherent sheaves, resp vector bundles) and the category of triples of such sheaves on $U$, such sheaves on $\widehat{X}$, and an isomorphism on $\widehat{X} - \Gamma\_x$. 


This is just because a section of a relative smooth curve is a Cartier divisor.


